Post-Optimal Analysis in Linear Semi-Infinite Optimization

123

S P R I N G E R B R I E F S I N O P T I M I Z AT I O N

Miguel A. GobernaMarco A. López

Post-Optimal Analysis in Linear Semi-Infi nite Optimization

SpringerBriefs in Optimization

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Miguel A. Goberna • Marco A. López

Post-Optimal Analysisin Linear Semi-InfiniteOptimization

123

Miguel A. GobernaStatistics and Operations ResearchUniversity of AlicanteAlicante, Spain

Marco A. LópezStatistics and Operations ResearchUniversity of AlicanteAlicante, Spain

ISSN 2190-8354 ISSN 2191-575X (electronic)ISBN 978-1-4899-8043-4 ISBN 978-1-4899-8044-1 (eBook)DOI 10.1007/978-1-4899-8044-1Springer New York Heidelberg Dordrecht London

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Preface

Linear semi-infinite optimization (LSIO) deals with linear optimization problemsin which either the dimension of the decision space or the number of constraints(but not both) is infinite. A typical feature of this type of optimization problems isthat boundedness (i.e., finiteness of the optimal value) does not imply solvability(i.e., existence of an optimal solution). In most LSIO applications, the data definingthe nominal problem are uncertain, so that the user must choose among differentuncertainty models, e.g., robust models, parametric models, probabilistic models, orfuzzy models, by taking into consideration the nature of the data, the computationaleffort required to solve the auxiliary problems, the available hardware and software,etc. Parametric models are based on embedding the nominal problem into asuitable topological space of admissible perturbed problems, the so-called spaceof parameters. Sensitivity analysis provides estimations of the impact of a givenperturbation of the nominal problem on the optimal value. Qualitative stabilityanalysis provides conditions under which sufficiently small perturbations of thenominal problem provoke only small changes in the optimal value, the optimalset and the feasible set. Quantitative stability analysis, in turn, yields exact andapproximate distances, in the space of parameters, from the nominal problem toimportant families of problems (e.g., from a bounded problem to the solvable ones)and error bounds (of Lipschitz-type) which are related to the complexity analysis ofthe numerical methods.

This Springer Brief on post-optimal analysis in LSIO allows us to answer“what if” questions on the basis of stability and sensitivity results whose proofsare generally omitted while their use is illustrated by means of comments andsuitable examples. It is intended as a guide for further readings addressed tograduate and postgraduate students of mathematics interested in optimization andalso to researchers specialized in parametric optimization and related topics (e.g.,algorithmic complexity). Moreover, it could be a useful tool for researchers workingin those fields where LSIO models arise in a natural way in uncertain environments(e.g., engineering and finance).

The book is organized as follows. Chapter 1 recalls the necessary preliminarieson the theory and methods of LSIO which are presented in a detailed way in our

vii

viii Preface

monograph Linear Semi-Infinite Optimization [102], published in 1998, aggregatingsome concepts related to complementary solutions which are used in sensitivityanalysis and updating the brief review of numerical methods. In Chap. 2 we discussthe advantages and disadvantages of five different approaches to uncertain LSIOwhich are illustrated by means of the portfolio problem with uncertain returns.The remaining chapters describe the state of the art in those models which havea substantial presence in the LSIO literature: on the robust approach to LinearSemi-Infinite Optimization (Chap. 3), sensitivity analysis (Chap. 4), qualitative sta-bility analysis (Chap. 5), and quantitative stability analysis (Chap. 6). The materialreviewed in Chaps. 3, 4, and 6 has been published after 1998 while part of thecontent of Chap. 5 was already analyzed in detail in [102, Chaps. 6 and 10]. After theintroductory Chaps. 1 and 2, Chaps. 3–5 can be read independently, while Chap. 5contains the preliminaries of Chap. 6. The remarks at the end of each section reviewthe antecedents and extensions of the exposed results and methods, while the lastremark of each chapter describes some open problems.

The authors want to thank the coauthors of the many joint works mentionedin this book: J. Amaya, E. Anderson, A. Auslender, P. Bosch, M.J. Cánovas, A.Daniilidis, N. Dinh, A. Dontchev, A. Ferrer, V.E. Gayá, S. Gómez, F. Guerra, A.Hantoute, V. Jeyakumar, V. Jornet, D. Klatte, A. Kruger, M. Larriqueta, G.Y. Li,R. Lucchetti, J.E. Martínez-Legaz, J.A. Mira, B. Mordukhovich, J. Parra, M.M.L.Rodríguez, G. Still, T.Q. Son, T. Terlaky, M. Théra, M.I. Todorov, F.J. Toledo, G.Torregrosa, V.N. Vera de Serio, J. Vicente-Pérez, M. Volle, and C. Zalinescu. Fromall of them we have learnt much. Our special acknowledgment also to M.J. Cánovas,J. Parra, M.M.L. Rodríguez, M. Théra, F.J. Toledo, and E. Vercher for their support,careful reading of the manuscript, and suggestions for improvement, to our studentsof the Degree of Mathematics in Alicante A. Navarro and R. Campoy for havingdrawn some figures, and to the participants in a doctoral course based on the drafttaught by one of the authors at Universidad Nacional de San Luis (Argentina),April 2013, whose comments and criticisms helped us to improve the quality ofthe manuscript.

Alicante, Spain Miguel A. GobernaMarco A. LópezOctober 2013

Contents

1 Preliminaries on Linear Semi-infinite Optimization . . . . . . . . . . . . . . . . . . . . . 11.1 Optimality and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Grid Discretization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.3.2 Central Cutting Plane Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.3 Reduction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.3.4 Feasible Point Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Modeling Uncertain Linear Semi-infinite OptimizationProblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.1 Five Paradigms to Treat Uncertain LSIO Problems . . . . . . . . . . . . . . . . . . . 24

2.1.1 The Stochastic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.2 The Fuzzy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.1.3 The Interval Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.4 The Robust Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1.5 The Parametric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Modeling Uncertain Portfolio Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Robust Linear Semi-infinite Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Uncertain Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Uncertain Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.1 Perturbing the Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Perturbing the RHS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Perturbing the Objective Function and the RHS. . . . . . . . . . . . . . . . . . . . . . . 59

5 Qualitative Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.1 Irrestricted Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Stability Restricted to the Domain of the Feasible Set . . . . . . . . . . . . . . . . 735.3 Well and Ill-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

ix

x Contents

6 Quantitative Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.1 Quantitative Stability of Set-Valued Mappings . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Quantitative Stability of the Feasible Set Mapping. . . . . . . . . . . . . . . . . . . . 85

6.2.1 Distance to Ill-Posedness with Respect to Consistency . . . . . . . 866.2.2 Pseudo-Lipschitz Property of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.2.3 Calmness of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Quantitative Stability of the Optimal Set Mapping . . . . . . . . . . . . . . . . . . . . 966.3.1 Distance to the Ill-Posedness with Respect to Solvability . . . . 966.3.2 Pseudo-Lipschitz Property of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 976.3.3 Calmness of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1Preliminaries on Linear Semi-infiniteOptimization

1.1 Optimality and Uniqueness

Ordinary (or finite) linear optimization, linear infinite optimization, and linearsemi-infinite optimization (LO, LIO, and LSIO in short) deal with linear opti-mization problems, where the dimension of the decision space and the number ofconstraints are both finite, both infinite, and exactly one of them finite, respectively.With few exceptions (as some classical applications collected in [102, Chap. 2],among them a model developed by G. Dantzig in his Ph.D. Thesis on statisticalinference, started in 1936 and interrupted by World War II, or some recent work,as [190]), most LSIO problems arising in practice have finitely many decisionvariables, so that they can be expressed as

P W infx2Rn hc; xis.t. ha .t/ ; xi � b .t/ ; t 2 T; (1.1)

where h�; �i denotes the Euclidean scalar product in Rn, T is an infinite set and

the data form the triplet .c; a; b/ 2 Rn � .Rn/T � R

T . The LSIO problem P

is said to be continuous whenever T is a compact Hausdorff topological space,b 2 C .T / (the linear space of real-valued continuous functions on T ), anda D .a1 .�/ ; : : : ; an .�// 2 C .T /n. We also write the problem P in (1.1) in matrixform as follows:

P W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;(1.2)

where c0 denotes the transpose of the column vector c 2 Rn. We denote by

F D fx 2 Rn W a0

t x � bt ; t 2 T g the feasible set of P . Obviously, F is a closedconvex set (and, conversely, any closed convex set is the solution set of some linearsystem as a consequence of the separation theorem).

M.A. Goberna and M.A. López, Post-Optimal Analysis in Linear Semi-InfiniteOptimization, SpringerBriefs in Optimization, DOI 10.1007/978-1-4899-8044-1__1,© Miguel A. Goberna, Marco A. López 2014

1

2 1 Preliminaries on Linear Semi-infinite Optimization

Introducing the so-called marginal function g W Rn ! R D R[fC1g by

g .x/ WD supt2T

�bt � a0

t x�; (1.3)

F D fx 2 Rn W g .x/ � 0g and P becomes an ordinary optimization problem with

linear objective and a unique constraint:

P W infx2Rn c0xs.t. g .x/ � 0:

The graph, the epigraph, and the hypograph of g are

gphg WD f.x; g .x// W g .x/ 2 Rg ;epi g WD ˚.x; �/ 2 R

nC1 W g .x/ � �� ;and

hypo g WD ˚.x; �/ 2 RnC1 W g .x/ � �� ;

respectively. When F ¤ ;, g is a proper convex function. Then, the domain of g isthe nonempty convex set

domg WD fx 2 Rn W g .x/ < C1g :

The convexity of g entails the convexity of the sublevel sets of g, fx 2 Rn W

g .x/ � �g, � 2 R, i.e., the quasiconvexity of g.Unfortunately, the reformulation of P as an ordinary convex optimization

problem is only useful when P is continuous for two reasons which are relatedto the theoretical analysis and the numerical treatment of P W1. It is difficult to get geometric information on the feasible set F from g.2. The computation of the convex subdifferential of g at a point x 2 F ,

@g .x/ WD ˚u 2 Rn W g .y/ � g .x/C u0 .y � x/ 8y 2 R

n�;

is a hard work (see [128] and references therein). Notice that, when P iscontinuous, Valadier’s formula yields

@g .x/ WD conv˚�at W g .x/ D bt � a0

t x; t 2 T�:

Throughout this chapter P denotes a given LSIO problem with fixed data.We denote by v .P / WD infx2F c0x (with the convention that inf; D C1)and S WD fx 2 F W c0x D v .P /g the optimal value and the optimal set of P ,respectively. Obviously, F is a closed convex set and S is an exposed face ofthe feasible set F , but (in contrast with LO) we may have S D ; even thoughv .P / 2 R.

1.1 Optimality and Uniqueness 3

Let us introduce some basic notation. Following Kortanek [161], we denoteby R

.T / the linear space of generalized finite sequences, whose elements are thefunctions � 2 R

T that vanish everywhere on T except on a finite subset of T .The notation R

.T / for the space of dual variables has been standard in semi-infiniteprogramming for several decades and was exported to infinite programming in [71].Let RC and RCC (R� and R��) be the sets of nonnegative and positive (nonpositiveand negative, respectively) real numbers. The positive cone in R

.T / is denoted byR.T /C and the null element of RT by 0T .Let us introduce first the basic notation on sets used in this book. By

convX WD�P

t2T�txt W xt 2 X 8t 2 T; � 2 R

.T /C , and

P

t2T�t D 1

�;

coneX WD�P

t2T�txt W xt 2 X 8t 2 T , and � 2 R

.T /C�;

affX WD�P

t2T�txt W xt 2 X 8t 2 T; � 2 R

.T /, andP

t2T�t D 1

�;

and

spanX WD�P

t2T�txt W xt 2 X 8t 2 T , and � 2 R

.T /

�;

we denote the convex hull, the convex conical hull (with the origin), the affine hull,and the linear hull of a nonempty subset X of a linear space and by intX , bdX , andclX the interior, the boundary, and the closure of a subset X of a topological space.By definition, all the algebraic hulls are empty whenever X D ;, except the convexconical hull, which is the singleton set formed by the zero vector. We denote by k�k2,k�k1, and k�k1 the Euclidean, the `1, and the supremum norm in R

n, with closedunit balls B2, B1, and B1, and associated distances d2.�; �/, d1.�; �/, and d1.�; �/,respectively. The notation k�k is used for a general norm, whereas d.�; �/ denotes itsassociated distance. The zero vector of Rn is denoted by 0n.

When ; ¤ X � Rn, equipped with the topology induced by the Euclidean

norm, rintX and rbdX represent the relative interior and the relative boundaryof X (i.e., the interior and the boundary of X w.r.t. the topology induced byRn in affX ). If X is a closed convex set, the recession cone of X is 0CX WDfv 2 R

n W c C v 2 X 8c 2 Xg, and it coincides with the set of all the limits ofthe form limr!1 �rxr , where �r 2 RC, xr 2 X , r D 1; 2; : : : ; and �r # 0.Moreover, dimX WD dim affX represents the dimension of a convex set X ,Xı WD fy 2 R

n W x0y � 0 8x 2 Xg is the positive polar of a convex cone X , andX? WD Xı \ .�Xı/ is the orthogonal subspace to a linear subspace X . Thelinearity linX of a convex cone X is the greatest linear subspace contained in Xand X \ .linX/? is the pointed cone of X (for which 0n is an extreme point).


We associate with P , or with its constraint system � D fa0t x � bt ; t 2 T g, the

following sets:

• The convex hull of constraints data:

C WD conv f.at ; bt / ; t 2 T g � RnC1:

• The first moment cone:

M WD cone fat ; t 2 T g � Rn:

• The second moment cone;

N WD cone f.at ; bt / ; t 2 T g D RCC � RnC1:

• The characteristic cone:

K WD N C cone f.0n;�1/g � RnC1:

Obviously, M D ProjRn .N / D Proj

Rn .K/, where ProjRn W Rn � R ! R

n

denotes the projection mapping on Rn, i.e., Proj

Rn .x; xnC1/ D x. We say that � isinconsistent whenever F D ; and it is strongly inconsistent in the particular casethat some finite subsystem of � is inconsistent. The characteristic cone K and itsclosure clK (both expressed in terms of the data) capture all relevant information on� and F , respectively. Concerning the moment cones, N describes the consistencyof � while M describes the boundedness of F .

Theorem 1.1.1 (Existence). A system � is inconsistent if and only if .0n; 1/ 2 clK,and it is strongly inconsistent if and only if .0n; 1/ 2 K.

Theorem 1.1.1 remains true replacing K with N . It is not the case for the LSIOversion of the famous Farkas Lemma. Recall that an inequality w0x � � is said tobe consequence of a consistent system � whenever w0x � � for all x 2 F .

Theorem 1.1.2 (Non-homogeneous Farkas Lemma). Let � be a consistentlinear system. Then, a linear inequality w0x � � is consequence of � if and only if.w; �/ 2 clK.

Two consistent systems � D fa0t x � bt ; t 2 T g and Q� D fQa0

t x � Qbt ; t 2 T g areequivalent if they have the same set of solutions, i.e., if F D QF ; in other words,if they constitute two alternative representations of the same closed convex set F .According to Theorem 1.1.2, we have [102, Theorem 5.10]:

• � and Q� are equivalent if and only if clK D cl QK.

Moreover, as shown in [102, Chaps. 5 and 9], F ¤ ; is:

• bounded, .0n;�1/ 2 int clK D intK ,M D Rn.

• a polyhedral convex set, clK is polyhedral.


• an affine manifold, the pointed cone of clK is cone f.0n;�1/g.• full dimensional, clK is pointed.

Denote s .x; t/ WD ha .t/ ; xi � b .t/ ; t 2 T . The slack function at x 2 Rn,

s .x; �/, allows us to check its feasibility: x 2 F if and only if s .x; �/ is nonnegativeon T . The set of zeros of the slack function at x is the so-called set of activeconstraints (also called binding constraints) at x 2 R

n W

T .x/ WD ft 2 T W s .x; t/ D 0g :

So, computing T .x/ leads us to the problem of finding the optimal set of theunconstrained global minimization problem inft2T s .x; t/. If the slack function isnon-identically zero and all the coefficients in the constraint system � are analyticfunctions of the index t , then T .x/ is a finite set. When these coefficients arepolynomial, computing T .x/ consists of solving an algebraic equation.

The cone of feasible directions and the active cone at x 2 F are

D .F I x/ WD fd 2 Rn W 9� > 0; x C �d 2 F g

and

A .x/ WD cone fat ; t 2 T .x/g ;

respectively. If t 2 T .x/ and d 2 Rn satisfies x C �d 2 F for some � > 0, then

bt C a0t �d D a0

t .x C �d/ � bt , so that a0t d � 0. So, D .F I x/ � A .x/ı and since

clA .x/ D A .x/ıı (by the Farkas Lemma for cones), one has

A .x/ � clA .x/ � D .F I x/ı : (1.4)

We now introduce four constraint qualifications (one local and three global)which are useful in different frameworks (optimality, duality, stability). We say thatP (or � ) satisfies:

• the local Farkas–Minkowsky constraint qualification (LFMCQ in brief) at x 2 Fif every consequence of � binding at x is consequence of a finite subsystem of� or, equivalently (by Theorem 1.1.2), if D .F I x/ı � A .x/, which itself isequivalent by (1.4) to D .F I x/ı D A .x/ I

• the Farkas–Minkowsky constraint qualification (FMCQ) if every consequence of� is consequence of a finite subsystem or, equivalently (by Theorem 1.1.2), if Kis closed;

• the Slater constraint qualification (SCQ) if there exists Ox 2 Rn (called Slater

point) such that a0t Ox > bt for all t 2 T or, equivalently, if Ox 2 F and T . Ox/ D ;I

• the strong Slater constraint qualification (SSCQ) if there exists Ox 2 Rn (called

SS point) and " > 0 such that a0t Ox � bt C " for all t 2 T or, equivalently, if

v .PSS/ > 0, where


PSS W sup.x;y/2RnC1 y

s.t. a0t x � bt C y; t 2 T:

Observe that the LSIO problem PSS is continuous if and only if P is continuous,and that checking the condition v .PSS/ > 0 does not require to solve PSS untiloptimality. Notice also that replacing each constraint a0

t x � bt in � with theinfinitely many constraints ka0

t x � kbt � 1, k 2 N, one gets another linearrepresentation of F such that all points of F are SS with " D 1 (even the pointsof bdF ). The existence of linear systems such that every constraint is inactiveat any feasible solution is an inconvenient of LSIO in comparison with LO(feasible direction methods use T .xk/ at the current iterate xk). This undesirablesituation is not possible whenever � is continuous and does not contain the trivialinequality 00

nx � 0, in which case intF is the set of Slater points [102, Corollary5.9.1]. Obviously,

SSCQ) SCQ.

Moreover, if P continuous and satisfies SCQ, then the compact convex set

conv .C [ f.0n;�1/g/ D conv f.at ; bt / ; t 2 T I .0n;�1/g

does not contain 0nC1 and so K D RC conv .C [ f.0n;�1/g/ is closed.Therefore,

P continuousSCQ holds

�) FMCQ holds ) LFMCQ holds at any x 2 F:

If x 2 F satisfies the Karush–Kuhn–Tucker (KKT) condition

c 2 A .x/ ; (1.5)

then c 2 D .F I x/ı by (1.4), so that c0 .x � x/ � 0 for all x 2 F , i.e., x 2 S .Actually, c 2 D .F I x/ı , x 2 S .

We now assume the existence of � 2 R.T /C such that

�x; �

�2 R

n � R.T /C is a

saddle point of the Lagrange function of P ,

L .x; �/ WD c0x CX

t2T�t .bt � a0

t x/;

i.e., we assume that

L .x; �/ � L�x; �

�� L

�x; �

�for all .x; �/ 2 R

n � R.T /C : (1.6)


Given any s 2 T , the first inequality in (1.6), with �t D �t for all t ¤ s 2 T and�s D �s C 1, yields bs � a0

sx � 0, so that x 2 F . Moreover, given x 2 F , we have

c0x D L .x; 0T / � L�x; �

�� L

�x; �

�D c0x C

X

t2T�t .bt � a0

t x/ � c0x;

so that (1.6) also implies x 2 S .

Observe that the first inequality in (1.6) entailsP

t2T �t .bt �a0t x/ D L

�x; �

��

L .x; 0T / � 0, which together with x 2 F yields the complementarity condition

�t .bt � a0t x/ D 0 for all t 2 T: (1.7)

Consider a LSIO problem P such that x 2 S ¤ F , which entails c ¤ 0n.Replacing each constraint a0

t x � bt with ka0t x � kbt � 1, k 2 N, we get another

LSIO problem with the same feasible set F and cost vector c, so that the optimal setis still S and any feasible point is SS. For the sake of simplicity we assume that thisis the case for the initial problem P . Then, .1.5) ) c D 0n (contradiction) while

(1.6) ) (1.7) ) � D 0T ) L��; ��D hc; �i and, from the second inequality

in (1.6), c0x � c0x for all x 2 Rn, i.e., c 2 .Rn/ı D f0ng, which also implies

c D 0n. Hence, the conditions (1.5) and (1.6) are sufficient, but not necessary, forthe optimality of x unless certain CQ holds.

Theorem 1.1.3 (Optimality). If the LFMCQ holds at x 2 F , then the followingstatements are equivalent to each other:

(i) x 2 S .(ii) c 2 A .x/.

(iii) There exists � 2 R.T /C such that

�x; �

�is a saddle point of L.

If the characteristic cone K captures the relevant information on the feasible setF of a consistent problem, a similar role plays the first moment cone regardingthe optimal set S . Indeed, S is bounded if and only if c 2 intM (see, e.g., [102,Corollary 9.3.1]).

The uniqueness of the optimal solution is a useful property in LO as it allows usto apply the classical sensitivity analysis results. A stronger property plays a similarrole in LSIO: an element x 2 F is a strongly unique solution of P if there exists˛ > 0 such that

c0x � c0x C ˛ kx � xk2 for all x 2 F: (1.8)

Obviously, (1.8) implies S D fxg.Theorem 1.1.4 (Uniqueness). x 2 F is a strongly unique solution of P if andonly if c 2 intD .F I x/ı.


So, given x 2 F , if c 2 intA .x/ then x is strongly unique, by (1.4), and theconverse statement holds if the LFMCQ holds at x.

The following simple example is used for illustrative purposes throughout thisbook. Due to the simplicity of its constraints, one can get an explicit (but ratherinvolved) expression of the corresponding marginal function g.

Example 1.1.1. Consider the continuous LSIO problem

P W infx2R2 c0xs.t. � .cos t / x1 � .sin t / x2 � �1; t 2

0; �

2

;

x1 � 0 .t D 2/; x2 � 0 .t D 3/;with different cost vectors c. We have F D ˚x 2 R

2C W kxk2 � 1�,

g .x/ D8<

:

max fkxk2 � 1;�x1;�x2g ; if x 2 R2C;

max fx1 � 1;�x2g ; if x … R2C and x2 � x1;

max fx2 � 1;�x1g ; if x … R2C and x2 > x1;

and

N D K D cone

8<

:�0

@cos tsin t1

1

A ; t 2h0;�

2

iI0

@1

0

0

1

A ;

0

@0

1

0

1

A

9=

;

(see Fig. 1.1), whose projection on R2 is M D R

2. Due to the closedness of K, Psatisfies the FMCQ and the LFMCQ at any feasible point. Moreover, Ox D �

12; 12

�is

a Slater point, so that the SCQ and the SSCQ hold too.

(a) If c D .1; 1/, S D ˚x1�

with x1 D 02 strongly unique. In fact, D�F I x1� D

A�x1� D R

2C, with c 2 intA .x/ D R2CC.

(b) If c D .�1;�1/, S D ˚x2�, with x2 D

�1p2; 1p

2

�not strongly unique. Here

D�F I x2� D ˚

x 2 R2 W x1 C x2 < 0

� [ f02g and A�x2� D RCc (Fig. 1.2

represents both cones translated to x2).(c) If c D .1; 0/, S D f0g � Œ0; 1�. Let x3 D .0; 1/ 2 S . Here D

�F I x3� D˚

x 2 R2 W x1 � 0; x2 < 0

� [ f02g and A�x3� D cone f.0;�1/ ; .1; 0/g.

Figure 1.3 represents the graph of the slack functions at xi ; s�xi ; ��,

i D 1; 2; 3.

The existence of strongly unique solution admits a characterization in terms ofthe relationship between c and K (see Corollary 4.1.1).

Remark 1.1.1 (Antecedents and Extensions). Proofs of Theorems 1.1.1 and 1.1.2can be found in [102, Theorem 4.4] and [102, Theorem 3.1], respectively. As shownin [71], both results are valid in convex infinite optimization (CIO, in brief).Theorems 1.1.3 and 1.1.4 were first proved in [199] (see also [102, Theorems 7.1and 10.5]). There exist many versions of the optimality theorem in CIO (see, e.g.,[71, 72, 175]).

1.2 Duality 9

Fig. 1.1 Characteristic cone

Fig. 1.2 Active and feasibledirections cones

1.2 Duality

If c 2 M there exists � 2 R.T /C such that

X

t2T �tat D c. Then, for any x 2 F ,one has

c0x DX

t2T �ta0t x �

X

t2T �tbt : (1.9)


Fig. 1.3 Slack functions of xi , i D 1, 2, 3

The Haar dual problem of P consists of maximizing the lower bound for c0xprovided by (1.9):

D W sup�2R.T /

C

X

t2T �tbts.t.

X

t2T �tat D c:

We denote by FD , SD , and v .D/ the feasible and optimal sets ofD, and its optimalvalue, respectively (with the convention that sup; D �1). Notice thatD is a LSIOproblem as it has finitely many constraints and infinitely many decision variables.By construction, the weak dual inequality v.D/ � v.P / always holds. Observe thatFD ¤ ; if and only if c 2M .

Other dual LSIO problems can be associated with P following general schemes.For instance, the Lagrangian dual problem of P is the unconstrained optimizationproblem

DL W sup�2R.T /

C

infx2Rn L .x; �/ :

Since

infx2Rn L .x; �/ D infx2Rn�P

t2T �tbt C˝c �Pt2T �tat ; x

˛�

D� P

t2T �tbt ; if � 2 FD;

�1; otherwise,

the optimal values and optimal sets ofD andDL coincide, i.e., v .D/ D v .DL/ andSD D SDL .

The pair P �D admits a geometric reformulation in terms of the characteristiccone K. In fact, on one hand, D consists of maximizing the last coordinate, xnC1,on the set

�P

t2T�t .at ; bt / ; � 2 R

.T /C�D cone f.at ; bt / ; t 2 T g D N;

1.2 Duality 11

or, equivalently, on the set K D N C .0n;�1/, under the constraint thatX

t2T �tat D c. So,

DG W supy2Rfy W .c; y/ 2 Kg

satisfies v .D/ D v .DG/ while SDG D fv .D/g whenever SD ¤ ;; i.e., K \˚.x; xnC1/ 2 R

nC1 W x D c� is a closed half-line. On the other hand, given ˛ 2 R,˛ � v.P / if and only if c0x � ˛ is consequence of � if and only if (by the FarkasLemma) .c; ˛/ 2 clK. Thus,

PG W supy2Rfy W .c; y/ 2 clKg

satisfies v .P / D v .PG/ and SPG D fv .P /g whenever v .P / 2 R, even thoughS D ;. Of course, the space of decisions is the real line R for both PG andDG. Theweak duality v.DG/ � v.PG/ is here consequence of the inclusion K � clK andthe optimal sets have at most one element.

The continuous dual problem of a continuous LSIO problem P is

DC W sup�2C0C.T /

RTbt d� .t/

s.t.RTatd� .t/ D c;

where C0C .T / represents the cone of nonnegative regular Borel measures on T .

Since R.T /C can be seen as the subset of C0C .T / formed by the nonnegative atomic

measures and P �DC satisfies the weak duality, v.D/ � v.DC/ � v.P /. Thus, anycondition guaranteeing a zero duality gap for the pair P �D guarantees also a zeroduality gap for P �DC (with attainment of the dual optimal value of DC wheneverv.D/ is attained). Thus, we shall consider the Haar dual problemD of P throughoutthis book.

The main LSIO duality theorems give conditions guaranteeing a zero duality gapwith attainment of either the dual or the primal optimal value when F ¤ ; ¤ FD .These situations are called strong (or infmax) duality and converse strong (orminsup/ duality, respectively. The infmax (or strong) duality theorem is a straight-forward consequence of the relationship between the pairs P �D and PG �DG.

Theorem 1.2.1 (Duality). Let F ¤ ; ¤ FD . Then the following statementshold:

(i) If K is closed, then v.P / D v.D/ 2 R and SD ¤ ;.(ii) If c 2 rintM , then v.P / D v.D/ 2 R and S is the sum of a nonempty compact

convex set with a linear subspace.

A feasible pair .x; �/ 2 F � FD is said to be a complementary solution of theprimal–dual pair P �D if

supp x \ supp� D ;;


where

supp x WD ˚t 2 T W a0t x > bt

�and supp� WD ft 2 T W �t > 0g

are called the support sets of x and �, respectively.It has been shown in [110] that a pair .x; �/ 2 F � FD is a complementary

solution of P �D if and only if v .D/ D v .P / and .x; �/ is a primal–dual optimalsolution, i.e., .x; �/ 2 S �SD . Moreover, given a point x 2 F , there exists � 2 FD

such that .x; �/ is a complementary solution of P �D if and only if x is an optimalsolution for some finite subproblem of P . A triplet .B;N;Z/ 2 �2T �3 is called anoptimal partition for P if there exists a complementary solution .x; �/ such thatB D supp x, N D supp�, and Z D TŸ .B [N/. Then, the nonempty elementsof .B;N;Z/ form a partition of T (a tripartition when the three sets are nonempty).A partition

�B;N ;Z

�is maximal if

B D[

x2Ssupp x; N D

[

�2SDsupp� and Z D T n .B [N/:

The uniqueness of the maximal partition is a straightforward consequence of

the definition. If there exists an optimal solution pair�x;�

�2S � SD such that

suppx D B and supp� D N , then the maximal partition is called the maximal

optimal partition. Hence, if S D fxg, SD Dn�o, and v .D/ D v .P /,

�supp x; supp�;TŸ

�supp x [ supp�

��is the maximal optimal partition.

If�B;N ;Z

�is an optimal partition such thatZ D ;, then it is a maximal optimal

partition. The maximal optimal partition may not exist.

Example 1.2.1. Consider the LSIO problem of Example 1.1.1:

P .c/ W infx2R2 c0xs.t. � .cos t / x1 � .sin t / x2 � �1; t 2

0; �

2

;

x1 � 0 .t D 2/; x2 � 0 .t D 3/;for three different choices of c 2 R

2. We have seen in Example 1.1.1 thatK is closedand M D R

2. So, by the duality theorem, v.P / D v.D/, with S ¤ ; compact andSD ¤ ; for any c 2 R

2.

(a) c D .1; 1/. We have S D f02g and, solving the system�P

t2T�t .at ; bt / D .c; v.D// ; � 2 R

.T /C�;

we get SD Dn�1o, with �12 D �13 D 1 and �1t D 0 for all t 2 0; �

2

. Since

supp 02 D0; �

2

and supp�1 D f2; 3g, �0; �

2

; f2; 3g ;;� is the maximal

optimal partition.

1.3 Numerical Methods 13

(b) c D .�1;�1/. Here S D ˚x2�, with x2 D

�1p2; 1p

2

�, so that

v.D/ D v.P / D �p2. The uniqueness of D follows from the relationshipbetween SD and SDG , and the identification of the characteristic cone K.In fact,

K \ .fcg � R/ Dn��1;�1;�p2 � �

�W � � 0

o;

and RC��1;�1;�p2

�is an extreme ray of K whose unique generator is

�a�4; b �

4

�. Thus, SD D

n�2o, with �2�

4D p2 and �2t D 0 for any t ¤ �

4

(observe that duplicating the constraint corresponding to t D �4

, nothingchanges in the primal problem but SD would be no longer a singleton). Sincesupp x2 D TŸ

˚�4

�and supp�2 D ˚

�4

�,�TŸ

˚�4

�;˚�4

�;;� is the maximal

optimal partition.

(c) c D .1; 0/. Now S D f0g � Œ0; 1� while SD Dn�3o, with �32 D 1

and �3t D 0 otherwise. Here supp�3 D f2g while, given x 2 S , we mayhave supp x D

0; �2

, supp x D

0; �2

[ f3g or supp x D 0; �

2

[ f3g.Thus, the optimal partitions are

�0; �

2

; f2g ; f3g�, �0; �

2

[ f3g ; f2g ;;�, and�0; �

2

[ f3g ; f2g ; ˚�2

��and

�0; �

2

[ f3g ; f2g ;;� turns out to be the maximaloptimal partition.

Remark 1.2.1 (Antecedents and Extensions). The seminal papers on Haar’s dualitywere published in the 1960s [59, 60]. As shown in [72], D and DL are alsoequivalent to the dual problem in Rockafellar’s sense [205], whose feasible setis R.T /.

Proofs of Theorem 1.2.1(i) and its extension to Lagrangian duality in CIO can befound in [102, Theorem 8.2] and [26], respectively. The interest by minsup dualityis quite recent. As observed in [107, Remark 6], Theorem 1.2.1(ii) can be provedfrom [102, Theorem 8.1]. An extension of Theorem 1.2.1(ii) to Lagrangian dualityin CIO has been proposed in [107].

1.3 Numerical Methods

We finish this introductory chapter with a brief description of some numericalmethods to solve the LSIO problem P in (1.1). The reader cannot expect efficientmethods allowing to solve any LSIO problem. Otherwise, we could computeefficiently the optimal value of any optimization problem

P1 W infx2X f .x/


by solving the equivalent LSIO problem

P2 W infy2R �ys.t. y � f .x/ ; x 2 X;

with v .P1/ D �v .P2/. More precisely, if x is an optimal solution of P1, then f .x/is an optimal solution of P2 and, conversely, if y is an optimal solution of P2, thenthe optimal set of P1 is the set of active indices of P2 at y D y.

The main drawback with the (linear and nonlinear) semi-infinite methods is thefact that checking the feasibility of a given x 2 R

n requires to compute the optimalvalue, v .Q .x//, of the so-called sublevel problem at x 2 R

n,

Q.x/ W inft2T s .x; t/ D inf

t2T fha .t/ ; xi � b .t/g ;

which is a global optimization problem. Even more, some algorithms require thecomputation of the set T .x/ of global minima of Q.x/, and this is only possibleunder strong assumptions on s and T , e.g., that s.x; t/ be a polynomial function oft and T is a finite dimensional interval.

1.3.1 Grid Discretization Methods

Discretization methods generate sequences of points in Rn converging to an optimal

solution of P by solving a sequence of LO problems. These problems are usuallysubproblems of P of the form

P .Tk/ W minx2Rn c

0x s.t. a0t x � bt for all t 2 Tk;

where Tk is a nonempty finite subset of T for k D 1; 2; : : : Take a fixed small scalar" > 0 (called accuracy) in order to guarantee finite termination.

Step k W Let Tk be given.

1. Compute a solution xk of P .Tk/.2. Stop if xk is feasible within the fixed accuracy ", i.e., a0

t xk � bt � " for all t 2 T .Otherwise, replace Tk with a new grid TkC1.

Obviously, xk is unfeasible before optimality. Grid discretization methods selecta priori sequences of grids .Tk/

1kD0 (usually satisfying Tk � TkC1 for all k). The

alternative discretization approaches generate the sequence .Tk/1kD0 inductively. For

instance, the classical Kelley cutting plane approach consists of taking TkC1 D Tk[ftkg, for some tk 2 T (as in Fig. 1.4), or TkC1 D .Tk [ ftkg/Ÿ

˚t 0k�

for somet 0k 2 Tk (if an elimination rule is included).

Convergence of discretization methods requires P to be continuous. Thisguarantees the convergence in the case of the Kelley cutting plane method while


Fig. 1.4 Feasibility cut

grid discretization methods require in addition the following density assumption onP (actually on T ): T is the union of a finite set U with another compact set V suchthat V D cl.intV /. In that case, the sequence of grids should satisfy U � Tk forall k.

Concerning the latter requirement, consider the problems in Example 1.1.1,which are continuous and T D

0; �2

[ f2; 3g satisfies the density assumptionwith U D f2; 3g and V D

0; �2

. If Tk is a grid in T such that U ¤ Tk , then the

feasible set F recedes in at least one of the directions .�1; 0/ or .0;�1/ and P .Tk/is unbounded in case (a), so that the sequence .xk/

1kD0 may not exist. In the contrary,

requiring U � Tk for all k, we have xk D 02 for k and 02 D limk!1 xk 2 S .The main drawbacks with these methods are undesirable jamming in the prox-

imity of S (unless P has a strongly unique optimal solution) and the increasing sizeof the auxiliary problems P .Tk/ (unless elimination rules are implemented). Thesemethods are only efficient for low-dimensional index sets, i.e., T is contained insome Euclidean space and dim affT � 3 (otherwise the cardinality jTkj of Tk growsvery fast with k). For instance, if T DQm

iD1 Œ˛i ; ˇi � and Tk is formed by successivebipartitions of the intervals Œ˛i ; ˇi �, i D 1; : : : ; m, then jTkC1j D

�2k C 1�m > 2km

for k D 1; 2; : : : For more details see [102, Chap. 11] and [178], and referencestherein.

1.3.2 Central Cutting Plane Methods

Central cutting plane methods start each step with a polytope containing a sublevelset of P and calculate a certain center of this polytope. The polytope is thenupdated by aggregating to its defining system either a feasibility cut (if the center isunfeasible) or an objective cut (otherwise). Let " > 0 be a fixed accuracy.


Fig. 1.5 Objective cut

Step k W Let Qk be a polytope containing some sublevel set of P .

1. Compute a center xk of Qk .2. If xk … F , set QkC1 D fx 2 Qk W a0

t x � btg, where t 2 T satisfies a0t xk < bt ,

and k D k C 1. Otherwise, continue.3. If c0xk � minx2Qk

c0xC ", stop. Otherwise, setQkC1 D fx 2 Qk W c0x � c0xkgand k D k C 1.

Concerning item 1, an obvious condition for the existence of Q0 is the bounded-ness of S . Now we assume thatQk D

˚x 2 R

n W c0i x � di ; i 2 I

�, with I finite and

ci ¤ 0n for all i 2 I . If one chooses the geometric center (as in [21]), the radiusof the greatest ball for the norm k�k2 contained inQk is maxx2Qk

mini2I d2 .x;Hi /,

whereHi WD˚x 2 R

n W c0i x D di

�. Since d2 .x;Hi / D c0

i x�dikcik2 ; i 2 I , this auxiliary

problem is equivalent to minx2Qkmaxi2I �d2 .x;Hi / and also to the LO problem

min.x;y/2RnC1 y

s.t. c0i x � di C kcik2 y � 0; i 2 I;c0i x � di ; i 2 I:

Alternatively, if one chooses the analytic center (as in [192]), the auxiliary problemconsists of computing a global minimizer of infx2Rn f .x/, where f is a barrierfunction for Qk (i.e., a function f such that f .x/ ! C1 as x ! bdQk), e.g.,the logarithmic barrier function f .x/ D �Pi2I log

�c0i x � di

�, if x 2 intQk , and

f .x/ D C1, otherwise.Item 2 aggregates to the current polytope Qk a feasibility cut (some constraint

of P violated by xk) when xk is unfeasible, whereas item 3 checks the "-optimalityof xk , aggregating to Qk an objective cut when the result is negative. Figure 1.5illustrates an objective cut.


Typically, these methods generate sequences of feasible and unfeasible pointswhich provide stopping rules for "-optimality and stop before optimality at afeasible solution. For instance, the method proposed in [65] generates at each stepa non-feasible point together with a feasible one, the result of shifting the currentunfeasible point toward a fixed Slater point with step length easily computable.

In particular, the so-called interior point constraint generation algorithm in [192],inspired in [182], updates the current discretization Pk (interpreted as a dualproblem of certain LO problem in standard format, say Dk) of P by selecting apoint in the vicinity of the central path of Pk and aggregating to the constraints ofPk some violated constraints; then the full dimension of the new feasible set FkC1is recovered and the central path is updated. This process continues until the barrierparameter is small enough, i.e., the duality gap approaches to zero. This algorithmdoes not generate feasible points of P (so that it is not an interior point method) butconverges to an "-optimal solution after a finite number of constraints is generated.Assuming that P is continuous and F is full dimensional and bounded, the authorsprovide complexity bounds on the number of Newton steps needed and on the totalnumber of constraints that is required for the overall algorithm to converge.

Discretization and cutting plane methods share the above-mentioned drawbacks.Convergence of central cutting plane methods requires continuity ofP together withthe boundedness of S , i.e., c 2 intM . Additionally, the interior point constraintgeneration algorithm requires the stronger assumption that F is a convex body (i.e.,a compact convex full dimensional set), i.e., the pointedness of clK (i.e., that clKhas no lines). Recall that the cones M and clK are defined in terms of the data. Allthese conditions hold in the problems considered in Example 1.1.1.

In conclusion, efficient implementations of the cutting plane methods turn outto be computationally faster than the discretization by grids counterparts, but theyrequire stronger assumptions.

1.3.3 Reduction Methods

Reduction methods, which were already known from Chebyshev approximation,replace P with a nonlinear system obtained from the optimality conditions.

Under suitable conditions (recall Theorem 1.1.3), if x is a minimizer of P , thereexist indices t j 2 T .x/, j D 1; : : : ; q .x/, with q .x/ 2 N depending on x, andnonnegative multipliers �j , j D 1; : : : ; q .x/, such that

c Dq.x/X

jD1�j a

�t j�:

We assume also the availability of a description of T � Rm as

T D ft 2 Rm W ui .t / � 0; i D 1; : : : ; mg ; (1.10)

where ui is smooth for all i D 1; : : : ; m.


Observe that q .x/ is the number of global minima of the sublevel problemQ.x/,provided that v .Q .x// D 0. In that case, t j 2 T .x/ if and only if t j is a minimizerof the (finite) sublevel problem at x

Q .x/ W inft2T ha .t/ ; xi � b .t/s.t. ui .t / � 0; i D 1; : : : ; m:

Then, under some constraint qualification, the classical KKT theorem yields the

existence of nonnegative multipliers �j

i , i D 1; : : : ; m, such that

˝r t a�t j�; x˛ � r t b

�t j� D

mX

iD1�j

i r tui�t j�

(1.11)

and

�j

i ui�t j� D 0; i D 1; : : : ; m:

In the typical case that T is an interval Œ˛; ˇ� � R, m D 2, u1 .t/ D t � ˛, u2 .t/ Dˇ � t , and (1.11) reads

*da�t j�

dt; x

+

� db�t j�

dtD

mX

iD1�j

i r tui�t j�:

Step k W Start with a given xk (not necessarily feasible).

1. Estimate q .xk/.2. Apply Nk steps of a quasi-Newton method (for finite systems of equations) to

8ˆˆˆ<

ˆˆˆ:

c Dq.xk/X

jD1�j a

�tj�

˝a�tj�; x˛ D b �tj

�; j D 1; : : : ; q .xk/

˝r t a�tj�; x˛ � r t b

�tj� D

mX

iD1�ji r tui

�tj�; j D 1; : : : ; q .xk/

�ji ui

�tj� D 0; i D 1; : : : ; m; j D 1; : : : ; q .xk/

9>>>>>>>>>=

>>>>>>>>>;

(1.12)

(with unknowns x, tj , �j , �ji , i D 1; : : : ; m, j D 1; : : : ; q .xk/), leading toiterates xk;l ; l D 1; : : : ; Nk .

3. Set xkC1 D xk;Nk and k D k C 1.

The reduction methods have two drawbacks: first, they require strong assump-tions on P (smoothness of the functions describing F and T ), the compactness ofT , the existence of some Slater point (so that P satisfies FMCQ); and, second, theyrequire sufficiently accurate approximate solution of (1.12) to start. The advantage


of reduction methods is their fast asymptotic behavior (as the quasi-Newton methodthey use). A reasonable strategy consists of combining, in a two-phase method,the advantages of discretization, which may provide an estimation of q .xk/ andan approximate solution of (1.12), and reduction, to improve this approximatesolution. Unfortunately, no theoretical result supports the decision to go from phase1 (discretization) to phase 2 (reduction) and, in practice, it is difficult to compute asuitable starting point for phase 2.

Example 1.3.1. Let us apply the above methods to the simple LSIO problem ofExample 1.1.1(b):

P W infx2R2 �x 1 � x2s.t. � .cos t / x1 � .sin t / x2 � �1; t 2

0; �

2

;

x1 � 0 .t D 2/; x2 � 0 .t D 3/:

Take the sequence of grids Tk D f2; 3g[˚�i4kW i D 0; : : : ; 2k�, k 2 N. Then �

42 Tk

for all k 2 N and the unique optimal solution of P .Tk/ is xk D�

1p2; 1p

2

�, with

a0t xk � bt � " for all t 2 T . Thus, the discretization algorithm, with an arbitrary

tolerance " > 0, stops at step 1 with x1 2 S . If, alternatively, we take Tk D f2; 3g [˚�i

4kC2 W i D 0; : : : ; 2k�, k 2 N, the sequence of optimal solutions generated by the

grid discretization method is

xk D�

sin

��4k C 18k C 4

�

��1 �1p2;1p2

!�1p2;1p2

2 S;

with kxkk2 > 1 for all k, so that the algorithm terminates after a finite number ofsteps with some unfeasible point.

Now we apply the geometric central cutting plane method. Taking Q0 DŒ0; 1�2 � F , x0 D

�12; 12

� 2 F . So the 1st cut is an objective one and Q1 D˚x 2 R

2 W x1 C x2 � 1; x1 � 0; x2 � 0�, whose geometric center is its incenter

x1 D p

2C 1p2C 2 ;

p2C 1p2C 2

!

D�1p2;1p2

2 S;

and the algorithm terminates in one step independently of the chosen tolerance.If one applies the analytic central cutting plane method with logarithmic

barrier function and the same initial polytope as above, Q0 D Œ0; 1�2, x0and Q1 are the same as before, but now x1 D

�23; 23

� 2 F . Then, Q2 D˚x 2 R

2 W x1 C x2 � 43; x1 � 0; x2 � 0

�, x2 D

�79; 79

� … F and so the nextcut is a feasibility cut. Since the deepest cut corresponds to t D �

4, Q3 Dn

x 2 R2 W p2 � x1 C x2 � 4

3; x1 � 0; x2 � 0

o, x3 D .0:68559; 0:68559/ 2 F ,

and so on. Observe that this algorithm generates feasible and unfeasible iterates sothat termination can be produced at a feasible approximate solution.


Concerning the reduction approach, we must represent T D 0; �

2

[ f2; 3g asin (1.10), e.g., T D ft 2 R W u .t/ � 0g with u .t/ D �t �t � �

2

�.t � 2/2 .t � 3/2,

and estimating the number of active indices at the minimum, in our case 1 or 2 (themaximum number of active indices at the boundary of F ). Taking q .x/ D 1, (1.12)becomes

8ˆ<

ˆ:

.1; 1/ D � .cos t; sin t /.cos t / x1 C .sin t / x2 D 1.sin t / x1 � .cos t / x2 D � du

dt

�u .t/ D 0

9>>=

>>;: (1.13)

It is easy to check that .x1; x2; t; �; �/ D�

1p2; 1p

2; �4; 2; 0

�is a solution of (1.13).

Newton and quasi-Newton methods provide sequences in R5 converging fast (at

least superlinearly) to this point provided an approximate solution is available(unfortunately, getting it is a difficult task!).

1.3.4 Feasible Point Methods

Feasible point methods generate sequences .xk/1kD1 of feasible points such that the

corresponding sequence of images by the objective function .c0xk/1kD1 are non-increasing. The main drawback with these methods is the computational effortrequired to find the optimal set of the sublevel problem at the current iterate xk :

Q.xk/ W mint2T s .xk; t/ D min

t2T fha .t/ ; xki � b .t/g :

In fact, the computation of all the global minima ofQ.xk/ is only possible wheneverT is a compact interval in R and a1 .�/ ; : : : ; an .�/ ; b .�/ 2 C1 .T / (the class ofanalytic functions on T , which contains the polynomial functions), in which case Pis said to be analytic. Classical feasible point methods generate a feasible directionat the current iterate xk by solving a certain LO problem, the next iterate beingthe result of performing a linear search in this direction improving the objective asmuch as possible until getting a point xkC1 2 bdF . The simplex-like algorithmof Anderson and Lewis [5] consists of alternating purification steps (providing anextreme point of F from the current iterate) and line search steps providing a pointof bdF . This method was adapted in [169] to problems which are analytic byblocks. The convergence of the sequences generated by simplex-like methods toan optimal solution of P is not guaranteed. Stein and Still [215] have proposedan interior point method for (not necessarily linear) semi-infinite programs whoseLSIO version requires the constraint system to be formed by subsystems of theform fha .t/ ; xi � b .t/ ; t 2 T g, where T is a convex subset of some vectorspace, and the function t 7! ha .t/ ; xi � b .t/ to be affine (a strong assumption).


More recently, Floudas and Stein [83] developed an efficient feasible point methodwhose underlying idea consists of replacing the hard global auxiliary problemQ.xk/ by a suitable convexification under assumptions weaker than those of [169](the coefficients of the inequalities of each block must be C2 instead of C1).

Remark 1.3.1 (Simplex-Like Methods). In the seminal papers [59, 60] it wasobserved that the extreme points of FD could be characterized in an algebraic way(as in LO). The corresponding simplex method for D was described in algebraicterms in [90] and in geometric terms in [98]. An extension to arbitrary infinitedimensional spaces has been proposed in [212].

Simplex methods for P are only possible under strong assumptions on theconstraints. A purification method for analytic LSIO problems was proposed in[5]. In the same paper was proposed the so-called hybrid method, which alternatespurification steps with linear search steps, providing infinite sequences in F withnon-increasing images by the objective function. A simplex method was proposedin [6] for a class of LSIO problems whose feasible set is quasipolyhedral (i.e., a setwhose intersection with polytopes is either empty or a polytope). The advantage ofsimplex-like methods are their generality (the dual simplex method does not requirecontinuity) and their common drawbacks are the lack of convergence theoremsverifiable in practice and the need of exact solutions of the subproblems (anunrealistic requirement).

In the absence of continuity, the LSIO problems could also be approximatelysolved by a suitable adaptation of the stochastic approach of [33].

Remark 1.3.2 (Available Solvers). The unique publicly available software for (lin-ear and nonlinear) semi-infinite optimization, NSIPS,1 uses the SIPAMPL softwarepackage, which extends AMPL environment to the SIPAMPL database (see [223]and the SIPAMPL manual2 for additional information). NSIPS is available on theNEOS server3 and includes four solvers: a discretization solver, a penalty solver, asequential quadratic programming solver, and an unfeasible quasi-Newton interiorpoint solver. Another family of publicly available solvers for (linear and convex)semi-infinite optimization is in preparation on the basis of the Remez penaltysmoothing algorithm in [8] and its implementation in [7]. Simple descriptionsof the most efficient LSIO methods (also for nonlinear semi-infinite optimizationproblems), together with their corresponding comments, have been uploaded toNEOS Optimization Guide4 by one of the authors.

Concerning commercial software, the Optimization Toolbox of Matlab version 2contains a solver for semi-infinite optimization with either T � R or T � R

2, called

1http://www.norg.uminho.pt/aivaz/nsips.html2http://plato.la.asu.edu/ftp/sipampl.pdf3http://www.neos-server.org/neos/4http://www.neos-guide.org/algorithms

http://www.norg.uminho.pt/aivaz/nsips.html

http://plato.la.asu.edu/ftp/sipampl.pdf

http://www.neos-server.org/neos/

http://www.neos-guide.org/algorithms


fseminf (seminf in the version 1.5), whose use is described at the Matlab Tutorial.5

The LSIO problems are solved by fseminf via discretization.

Remark 1.3.3 (LSIO Applications). LSIO has been used: first, as a conceptual toolin economic theory, games, or geometry; second, as a computational tool in func-tional approximation, robust statistics, or semidefinite optimization; and third, as amodeling tool for real problems arising in engineering, health care, or spectrometry.Many users have had difficulties with the latter type of applications due to the lack,until recently, of publicly available software (usually the authors have felt obliged toimplement standard or ad hoc numerical methods). We enumerate below some fieldswhere LSIO has been applied in at least one of the three ways. A large collection ofreferences, published before 2010, can be found in http://wwwhome.math.utwente.nl/~stillgj/sip/lit-sip.pdf.

• Environmental engineering [90, 102, 125, 126, 129, 142, 237].• Optimal design [155, 231].• Telecommunication networks [91].• Control problems [90, 91].• Economic theory [91].• Finance [65, 162, 167, 186, 196, 224].• Game theory [91].• Spectrometry [61].• Health care [192].• Probability and Statistics [2, 22, 23, 80, 81, 91, 102].• Machine Learning [18, 184, 185, 193, 194, 213, 217].• Data envelopment analysis [91, 102].• Functional approximation [90, 102, 132, 163, 218].• Computational linear algebra [102].• Linear functional equations [90, 102].• Convex geometry [91, 149, 198].• Location problems [102].• Robust optimization [91, 178].• Semidefinite optimization [91, 164].• Geometric optimization [102].• Combinatorial optimization [164].

Remark 1.3.4 (Two Open Problems in Deterministic LSIO).

1. Convergence theorems for simplex-like methods.2. Complexity analysis of LSIO methods.

5http://serdis.dis.ulpgc.es/~ii-its/MatDocen/laboratorio/manuales/OPTIM_TB.PDF

http://wwwhome.math.utwente.nl/~stillgj/sip/lit-sip.pdf.

http://wwwhome.math.utwente.nl/~stillgj/sip/lit-sip.pdf.

http://serdis.dis.ulpgc.es/~ii-its/MatDocen/laboratorio/manuales/OPTIM_TB.PDF

Chapter 2Modeling Uncertain Linear Semi-infiniteOptimization Problems

In most LSIO applications part of the data, if not all of them, are uncertain asa consequence of error measurements or estimations. This uncertainty is inherentto the data in fields as environmental engineering, telecommunications, finance,spectrometry, health care, statistics, machine learning, or data envelopment analysis,just to mention some applications listed in Remark 1.3.3. In this chapter we considergiven an uncertain LSIO problem

P0 W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;(2.1)

which is many times the result of perturbing the data of a nominal problem

P W infx2Rn c0xs.t. a0

t x � bt ; t 2 T:(2.2)

Nevertheless, most authors ignore this fact, limiting themselves to solve a particularinstance of P0 without further analysis. As uncertain optimization problems canonly be analyzed and/or solved in the framework of specific models, this chapteris intended to help potential users of LSIO to choose suitable models throughthe description and comparison of different alternatives and the discussion of asignificant uncertain LSIO problem. The next section introduces five differentparadigms to treat uncertainty: the stochastic, the fuzzy, the interval, the robust,and the parametric approaches. Almost all the existing literature on uncertain LSIOis focused on the latter two approaches.


23

24 2 Modeling Uncertain Linear Semi-infinite Optimization Problems

2.1 Five Paradigms to Treat Uncertain LSIO Problems

2.1.1 The Stochastic Approach

Assume that the uncertain data are random variables with a known probabilitydistribution. Each realization of these random variables provides a deterministicLSIO problem called scenario. Although it is impossible to obtain in practicethe probability distribution of the optimal value, its empirical distribution can beapproximated via simulation by solving a sample of scenarios.

Probabilistic models are a subclass of stochastic models consisting of optimiza-tion problems with a deterministic objective function and constraints involvingprobabilities of events which are expressed in terms of the random data. Fuzzymodels and interval models can be seen as variants of the stochastic models.

2.1.2 The Fuzzy Approach

Now we recall some standard concepts on single-valued functions which are used inthe fuzzy and the parametric approaches. Let X be a topological space and denoteby Nx the family of neighborhoods of x 2 X . Let f W X ! R be given. Recall thata function f which is finite-valued around x 2 X is continuous at that point if, foreach " > 0, there exists V 2 Nx such that f .x/ � " < f .x/ < f .x/C " for allx 2 V . This concept can be split into two weaker ones:

• f is lower semicontinuous at x 2 X (lsc in brief) if, for each � < f .x/ thereexists V 2 Nx such that � < f .x/ for all x 2 V I f is lsc if it is lsc at anyx 2 X .

• f is upper semicontinuous at x 2 X (usc) if, for each � > f .x/ there existsV 2 Nx such that � > f .x/ for all x 2 V I f is usc if it is usc at any x 2 X .

It is easy to prove that f is lsc if and only if epi f is closed. This is the caseof the marginal function g defined in (1.3) as epig is intersection of closed half-spaces. The lsc hull of f is the greatest lsc minorant of f , i.e., the function whoseepigraph is cl epif . Obviously, f is usc at x 2 X if and only if its opposite function�f is lsc at x. So, f is usc if and only if hypo f is closed and the usc hull of f(i.e., the smallest usc majorant of f ) is the function whose hypograph is cl hypo f .Moreover, if f is finite-valued, it is continuous if and only if epi f and hypo f areclosed, and this implies that gph f is closed. As the function f W R! R such that

f .x/ D�0; if x D 0;1x2; otherwise,

(2.3)

shows, the converse is not true unless f W Rn ! R is bounded.

2.1 Five Paradigms to Treat Uncertain LSIO Problems 25

Now we assume that X D Rn. Recall that f is concave (proper, quasiconcave)

when�f is convex (proper, quasiconvex, respectively), i.e., when hypo f is convex(hypo f is a nonempty set without vertical lines, the superlevel sets of f are convex,respectively). We define the domain of a concave function f as the domain ofthe convex function �f , i.e., the convex set dom f WD fx 2 R

n W f .x/ > �1g.Analogously, we define the concave subdifferential of f at x 2 R

n as the symmetricw.r.t. the origin of the convex subdifferential of the convex function �f , i.e., theclosed convex set

@f .x/ WD ˚u 2 Rn W f .x/ � f .x/C u0 .x � x/ 8x 2 R

n�:

A fuzzy set A defined in a topological space X (called universal set) ischaracterized by a usc function �A W X ! Œ0; 1� called membership function ofA whose images �A .x/ ; x 2 X , represent the grade of membership of x in A (�Acan be seen as the fuzzy counterpart of a probability distribution on X ). A fuzzynumber A is a fuzzy set defined in R whose membership function is quasiconcaveand satisfies supx2R �A .x/ D 1.

Denote by F the class of fuzzy numbers. The product ˛A of ˛ 2 R and A 2 Fis the fuzzy number corresponding to the membership function �˛A .x/ WD �A

�x˛

�,

if ˛ ¤ 0, and �0A is the characteristic function 0 of 0 (i.e., 0 .0/ D 1 and0 .x/ D 0 otherwise) while the sum ACB of A;B 2 F is the fuzzy number whosemembership function is

�ACB .x/ WD minxDuCv

f�A .u/ ; �B .v/g :

So, the linear combination a0x DnX

1D1aixi of a1; : : : ; an 2 F, with x1; : : : ; xn 2 R,

is well defined by induction. Observe that, by definition of fuzzy number, givenA 2 F and a level � 2 Œ0; 1�, the superlevel set fx 2 R W �A .x/ � �g is aclosed convex set; so, we can write fx 2 R W �A .x/ � �g D

Al�; A

u�

, where

Al�; Au� 2 R WD R[f˙1g. The fuzzy numbers can be compared in different

ways. The classical one consists of defining A B whenever Al� � Bl� and

Au� � Bu

� for all � 2 Œ0; 1�, but then is just a partial order on F. Total binaryrelations can be defined on F by means of centrality measures of the membershipfunctions called ranking functions. For instance, the Roubence ranking functionR .A/ D 1

2

R 10

�Al� C Au

�

�d� (an average of the centers of the superlevel sets of

�A) allows us to define A B whenever R .A/ � R .B/; sinceR .A˛/ D 0 for anyA˛ 2 F such that �A˛ .x/ D exp

��˛x2�, ˛ > 0, we conclude that the latter binaryrelation is reflexive and transitive, but not antisymmetric.

Thus, once selected a suitable binary relation on F, the linear inequality a0x b, where a1; : : : ; an; b 2 F, has a precise meaning. Now the problem is how toreplace a fuzzy inequality a0x b with a (possibly nonlinear) deterministic system(crisp system, in fuzzy terminology), taking into account that generally .AC B/l� ¤


.A/l� C .B/l�, .AC B/u� ¤ .A/u� C .B/u�, and R .AC B/ ¤ R .A/ C R .b/. Thiscan be done by defining an ad hoc membership function for a0x. The fuzzy approachsketched here interprets the uncertain scalar data inP0 as fuzzy numbers with knownmembership functions, and the task consists of computing a vector x 2 F (thesolution set of the fuzzy system fa0

t x bt ; t 2 T g) such that the fuzzy inequalityc0x c0x holds for all x 2 F . The latter fuzzy system is frequently inconsistentwhen one compares the fuzzy numbers by means of the partial order defined throughthe superlevel sets.

2.1.3 The Interval Approach

Assume that any instance of the uncertain scalar data in P0 takes values on agiven interval (this assumption is weaker than the uniform distribution on thecorresponding interval). In other words, each scalar data in P0 is interpreted as aninterval A˙ D Al ; Au

, Al ; Au 2 R.

Denote by I the class of compact intervals in R. We define the product of ˛ 2 R

times A˙ 2 I by ˛A˙ WD ˛Al ; ˛Au

, if ˛ � 0, and ˛A˙ WD

˛Au; ˛Al,

otherwise. Similarly, we define the sum of A˙; B˙ 2 I as A˙ C B˙ WDAl C Bl ; Au C Bu

. This way, given a1 ; : : : ; an 2 I and x1; : : : ; xn 2 R, the

linear combinationPn

iD1 xiai is a well-defined element of I. Defining A˙ B˙whenever Al � Bl and Au � Bu, defines a partial order on I, so thatthe inequalities

PniD1 xiai b˙ and

PniD1 xi ci

PniD1 xi ci have precise

meanings too.The task consists then of determining a vector x 2 F (the solution set of the

interval system˚Pn

iD1 xiait bt ; t 2 T�) such that

PniD1 xi ci

PniD1 xi ci

for all x 2 F . In this model the optimal valuePn

iD1 xi ci is also an interval.

2.1.4 The Robust Approach

This approach provides a deterministic framework for studying mathematicalprogramming problems under uncertainty. Robust optimization models are basedon the description of uncertainty via sets, as opposed to probability distributions(membership functions, intervals) which are used in stochastic (fuzzy, interval)approaches.

Robust models assume that all instances of the data belong to prescribed sets(not necessarily intervals or boxes), but now the task consists of minimizing theworst possible value of the objective function on the set of points which are feasiblefor any possible instance of the constraints. Indeed, when only the cost vector c isuncertain in P0, and its uncertainty set is C � R

n, then the robust model replacesthe task “inf c0x” in P0 with “inf supc2C c0x”; alternatively, when the constraint


corresponding to index t 2 T is uncertain, with uncertainty set Ut � RnC1, then

the robust model replaces a0t x � bt in P0 with the linear semi-infinite system

fa0x � b; .a; b/ 2 Utg I consequently, when the uncertainty affects both costs andconstraints, defining Ut WD f.at ; bt /g whenever the constraint a0

t x � bt is certain,risk-averse decision makers prefer the (pure) robust optimization model

PR W infx2Rn supc2C c0xs.t. a0x � b; .a; b/ 2 S

t2TUt ;

to stochastic, fuzzy, interval, and parametric models. What the users expect from arobust optimization model is numerical tractability and existence of some optimisticcounterpart (concept that we define in a precise way in Chap. 3), which is obtainedvia duality, both problems having the same optimal value. The adjective “tractable”means in (finite) linear and convex programming that there exists an equivalentproblem for which there are known solution algorithms with worst-case runningtime polynomial in a properly defined input size [20]. Concerning uncertain LSIO,as this type of optimization problems are almost always hard, “tractable” meansin this Brief that there exists some algorithm providing an "-optimal solution in“reasonable time” for any " > 0.

Decision makers less risk-averse may prefer a mixed model which combines,e.g., the robust approach w.r.t. the constraints with the probabilistic (fuzzy, interval,parametric) one w.r.t. the objective function. These mixed models are obtained fromP0 in (2.1) by interpreting c as a random (fuzzy, interval, parametric) vector. Letus mention the existence of a stream of works comparing probabilistic and robustmodels for certain types of finite uncertain optimization problems with random data,whose main aim consists of guaranteeing that, under suitable assumptions on theuncertain sets, any robust feasible solution satisfies the probabilistic constraints withhigh probability (see [20] and references therein).

2.1.5 The Parametric Approach

Parametric models are based on embedding the nominal problem P in (2.2), identi-fied with the triplet � D .c; a; b/, into a suitable topological space of admissibleperturbed problems, the so-called space of parameters ˘ . The topology on ˘usually corresponds to some measure of the size of the admissible perturbations.When the perturbations are required to preserve the number n of decision variablesand the index set T , there is a consensus about the convenience of measuring thedistance between two parameters �1 D

�c1; a1; b1

� 2 ˘ and �2 D�a2; b2; c2

� 2˘ by the box extended distance

d .�1; �2/ W D max˚��c1 � c2�� ; supt2T

��a1t ; b1t

� � �a2t ; b2t��

D max˚��c1 � c2�� ; d .�1; �2/

�; (2.4)


with �1; �2 being the constraint systems of �1; �2, respectively, and where k�k isany norm in R

n and in RnC1. A relevant particular case is

d1 .�1; �2/ D max˚��c1 � c2��1 ; supt2T

��a1t ; b

1t

� � �a2t ; b2t��1

�; (2.5)

and in a similar way we shall define d2 .�1; �2/.These extended distances describe the topology of the uniform convergence.

When all the data in the nominal problemP , represented by the triplet � D .c; a; b/,can be perturbed, the parameter space is˘ D .Rn/T �RT �Rn; in the general case,and ˘ D C .T /n � C .T / � R

n, in the continuous case. Observe that ˘ is a reallinear space for the componentwise operations and it is not connected in the generalcase as the sets of parameters with bounded and unbounded constraints data sets arecomplementary open cones.

Qualitative stability analysis provides conditions under which sufficiently smallperturbations of the nominal problem provoke small changes in the optimal value,the optimal set, and the feasible set. The (primal) optimal value function is thesingle-valued extended function # W ˘ ! R such that # .�/ is the optimal valueof P (i.e., # .�/ D v .P /), whose desirable stability properties are the lower andupper semicontinuity. The optimal set and the feasible set mappings are set-valuedmappings. The (primal) feasible set mapping F W ˘ � R

n associates with each� 2 ˘ the feasible set F .�/ of P (the LSIO problem associated with �) while the(primal) optimal set mapping S W ˘ � R

n associates with each � 2 ˘ the optimalset S .�/ of P . In a similar way we shall consider the dual optimal value function#D W ˘ ! R, the dual feasible set mapping FD W ˘ � R

.T /C and the dual optimal

set mapping SD W ˘ � R.T /C assigning to � 2 ˘ the optimal value, the feasible set

and the optimal set of D, respectively. Observe that # (and also #D) is positivelyhomogeneous, i.e., # .��/ D �# .�/ for any � > 0 and � 2 ˘ .

At this point we must recall the basic continuity concepts for set-valued mappings(as F and S). Consider two topological spaces Y and X (in our parametricframework, X D R

n is the space of decisions of the nominal problem while Yis some topological subspace of the space of parameters ˘ ). Consider a set-valuedmapping M between Y andX , i.e., M W Y � X , and y 2 Y such that M .y/ ¤ ;.Then :

• M is (Berge-Kuratowski) lower semicontinuous (lsc, in brief) at y if for eachopen set V � X verifying M .y/ \ V ¤ ;, there exists U 2 Ny such thatM .y/ \ V ¤ ;, for all y 2 U .

• M is (Berge-Kuratowski) upper semicontinuous (usc, shortly) at y if for eachopen set V � X verifying M .y/ � V , there exists U 2 Ny such that M .y/ �V , for all y 2 U .

• If Y and X are pseudometric spaces, M is said to be closed at y if for allsequences .yr /1rD1 � Y and .xr /1rD1 � X satisfying xr 2M.yr / for all r 2 N,limr!1 yr D y and limr!1 xr D x, one has x 2M.y/.


Fig. 2.1 Lower and upper semicontinuity

Let us observe that if M is a single-valued mapping, lower and upper semicon-tinuity coincide with the ordinary notion of continuity of M W Y ! X . Continuityalso implies closedness, but the converse is not true (recall the function f in (2.3)).

The graph and the domain of M are

gphM WD f.y; x/ 2 Y �X W x 2M .y/g

and

domM WD fy 2 Y WM .y/ ¤ ;g D ProjY gphM;

respectively.Lower semicontinuity precludes the image set M .y/ to shrink drastically for

y close to y, whereas upper semicontinuity avoids the opposite situation, in otherwords, that the image sets explode in size around y. Closedness of M at y meansthat gphM contains the limits of its sequences whose images by ProjY gphMconverge to y. In Fig. 2.1 the graph of a set-valued mapping M W R � R is theblue shaded area, and we show three different situations at the points y (usc, notlsc), Qy (lsc, not usc), and Oy (not lsc, not usc). Moreover, M is non-closed at Qyand Oy.

Closedness of M at every y 2 domM is equivalent to the closedness of gphMin the product space Y �X . Closedness and upper semicontinuity compete to be thecounterpart concept of the lower semicontinuity.

It makes sense to consider that the feasible set mapping F is also defined onthe space of systems associated with the parameters � D .a; b/ (these systemsare also called semi-infinite and appear in different fields). In this way we may use


indistinctly F.�/ or F.�/. Similarly for the dual feasible mapping, i.e., FD.�/ isthe same thing that FD.�D/ where �D WD fPt2T �tat D cg. Observe that F is

closed for free. Indeed, let � D�a; b

�2 , and two sequences .�r /1rD1 �

and .xr /1rD1 � Rn satisfying xr 2 F.�r/ for all r 2 N, limr!1 �r D � and

limr!1 xr D x. Let �r D fhar .t/ ; xi � br .t/ ; t 2 T g, r 2 N. Fixed t 2 T ,taking limits as r !1 in har .t/ ; xri � br .t/, one has ha .t/ ; xi � b .t/ : Hence,x 2 F.�/ and so F is closed at � . We conclude that, in the framework of thestability of the feasible set, closedness is too weak while upper semicontinuity istoo strong (it hardly holds when the image is non-compact). Concerning S , we willsee in Sect. 5.1 that its closedness coincides essentially with the lsc property of F .The stability of FD W ˘ � R

.T /C has only been analyzed by assuming that R.T / is

equipped with the k�k1 and k�k1 norms [105] while the stability of SD has not beenconsidered yet.

Quantitative stability analysis deals with error bounds on distances in thedecision and the parameter spaces. A fundamental formula obtained by Hoffmanin 1952 [135] provides an error bound on the distance from any point of R

n tothe solution set of a linear system in terms of the most violated constraint. Thisformula turned out to be related to the complexity of numerical methods in LO. Forthis reason, many extensions and variations of this result have been proposed fordifferent types of systems, including those arising in LSIO. The parametric modelsalso allow to compute the distance to ill-posedness relative to some property enjoyedby the nominal problem, i.e., the minimum size of those perturbations of P whichprovide perturbed problems not enjoying such a property.

Sensitivity analysis provides estimations of the impact of a given perturbationon the optimal value. Most of the existing literature on sensitivity analysis in LOhas focused on determining linearity (or affinity) regions of the optimal valuefunction under perturbations of the vector cost c, or perturbations of the right-hand side (RHS) vector b, or both (it is difficult to study the effect on the optimalvalue of perturbations of the left-hand side a even in LO). Sensitivity analysis inLO can be approached from three different perspectives. The classical approachis based on the use of optimal basis (the available information when the simplexmethod attains an optimal extreme point of the polyhedral feasible set). Thisapproach cannot be extended to LSIO because the number of active constraintsat an extreme point of the feasible set is seldom more than one (in particular, thesmooth convex bodies as Euclidean balls and ellipsoids have a unique supportinghyperplane at every boundary point). The duality approach provides conditionsfor the affinity on segments or half-lines of the optimal value function for non-simultaneous perturbations of costs and RHS. The third approach to sensitivityanalysis in LO exploits the optimal partitions computed by the interior point methodwhen optimality is achieved, allowing to treat simultaneous perturbations of both,costs and RHS.

Example 2.1.1. Consider the LSIO problem, say P , of Example 1.1.1(c) andassume that the RHS of the subsystem corresponding to indexes t 2 0; �

2

can


be perturbed by a common parameter y 2 R (to be precise, the parameter space isthe set of triplets .c; a; b/, where bt D �y for all t 2 0; �

2

, and it can be identified

with R). Then we have to consider the parametric LSIO problem

P .y/ W infx2R2 x1s.t. � .cos t / x1 � .sin t / x2 � �y; t 2

0; �

2

;

x1 � 0 .t D 2/; x2 � 0 .t D 3/:

Observe that P .1/ P . Here F , S W R � R2 are F .y/ D ˚

x 2 R2C W kxk2 � y

�

and S .y/ D F .y/ \ .f0g � R/ while # .y/ D 0 if y 2 RC and # .y/ DC1 otherwise. So, we have domF D domS D dom# D RC, gph# DRC � f0g, gphF D ˚

.y; x1; x2/ 2 R3C W kxk2 � y

�(see Fig. 2.2) and gphS D˚

.y; 0; x2/ 2 R3C W 0 � x2 � y

�(see Fig. 2.3).

Since epi# is closed while epi .�#/ is not, # is lsc but not usc (the latter propertyfails at y D 0). Concerning F and S , the situation is exactly the opposite: they areusc but not lsc. In fact, the latter property fails at y D 0 as S .0/ D F .0/ D f02gwhile S .y/ D F .y/ D ; for all y < 0. Moreover, S is closed due to the closednessof gphS (recall that F is always closed).

Concerning the stability of P , since the three mappings are lsc and usc at y D1, and F and S are also closed at that parameter, we conclude that the nominalparameter is stable in all senses defined up to now (other stability concepts will beintroduced later).

Observe that the distance from P (or the corresponding parameter y D 1) toinconsistency is 1 because F .y/ ¤ ; for all y 2 R such that d1 .y; 1/ Djy � 1j � 1 while F �� 1

r

� D ; for all r 2 N, with d1�� 1

r; 1� D 1 C 1

r! 1.

The problem P .0/ is ill-posed in the consistency sense as any neighborhood of 0contains both consistent and inconsistent problems. Since domS D dom# , we canreplace “consistency” by “solvability” in the last two paragraphs.

Fig. 2.2 Graph of thefeasible set mapping


Fig. 2.3 Graph of theoptimal set mapping

Finally, notice that the sensitivity analysis is trivial in this example since # isconstant on RC (i.e., changes smaller than 1 in the nominal problem have no effecton #).

The parametric model most frequently encountered in the recent literature onstability in LSIO considers perturbations of all data. One reason for this is that thecharacterizations of different qualitative stability properties in this model becomesufficient conditions for the remaining models and sometimes these conditions arealso necessary. Analogously, the formulae providing the distance to ill-posednessare at least upper bounds in other models whereas the error bound are still valid(although they could be improved). Sometimes it is more difficult to study thestability under perturbations of part of the data (usually the LHS a) than underthe perturbations of all data while the continuous models allow to use useful topo-logical and analytical tools as the Urisohn’s lemma or Robinson–Ursescu theorem(provided that gphF is convex, as it is whenever a is fixed). Few sensitivity analysisresults have been published on LO under perturbations of data including a, so thatthe reader only can expect sensitivity analysis results on LSIO under perturbationsof the remaining data; thus, Sects. 4.1, 4.2, and 4.3 consider perturbations of c,b, and the couple .b; c/, respectively. Simultaneous perturbations of .c; a; b/ areconsidered in Sects. 5.1, 5.3, 6.2.1, and 6.3.1, separate perturbations of a; b, and cin Sect. 5.2, perturbations of b and .a; b/ in Sect. 6.2.2, and perturbations of .c; b/in Sect. 6.3.2.

The convenience and viability of the alternative models sketched in this sectionfor a specific uncertain LSIO problem depend on the nature of its data, the attitudeof the decision maker toward risk, the tractability of the auxiliary problems tobe solved, and the availability of hardware and software facilities. In conclusion,instead of claiming the superiority of one of the above models on the others, theauthors agree with the famous statement of the statistician George Box: “all modelsare wrong, but some are useful” [28].

2.2 Modeling Uncertain Portfolio Selection 33

Remark 2.1.1 (Antecedents).The Stochastic Approach: The existing literature on probabilistic models for uncer-tain semi-infinite problems [67–69, 211] does not include contributions on LSIO.

The Fuzzy Approach: The first works on fuzzy optimization were published in the1970s [13, 216]. In particular, [235] showed the way to reduce fuzzy LO problemswith linear membership functions to ordinary LO problems. There exists a wideliterature on fuzzy optimization via ranking functions (see, e.g., [76, 79, 171, 228],and the review paper [32]). The survey paper [170] reformulates and solvesfuzzy LO problems with crisp objective function as LSIO ones, and the samemethodology could be applied to LSIO problems with fuzzy constraints. To theauthors knowledge, no work has been published on fuzzy LSIO. Other connectionsbetween fuzzy optimization and LSIO have been established in [138, 195], and[227], where convex fuzzy optimization problems, semi-infinite fuzzy systemswith concave membership functions (whose unique illustrative example is linear),and optimal design of alumina ceramics with fuzzy data, respectively, have beenreformulated as LSIO problems.

The Interval Approach: Bhattacharjee et al. [24] review the interval approachin nonlinear semi-infinite optimization without considering the particular case ofLSIO. The interval approach has been used to tackle uncertain LSIO problemsarising in environmental engineering in a rather empirical way [125, 126, 129, 130,142, 173, 237]. In the more complex interval model of [173] the decision variablesare replaced with decision intervals.

The Robust Approach: LO problems with uncertain constraints have been treatedin a robust way in [214] under the name of inexact linear programming, and theirrelationship with LSIO was explored in [219], whose authors consider uncertainconstraints (with coefficient vectors .at ; bt / typically ranging on some ball fora weighted supremum norm). The antecedents of robust LSIO are reviewed inChap. 3.

The Parametric Approach: The extended distance defined in (2.5) was introducedin the seminal paper [123]. The first results on the stability of F , S , and # werepublished in the 1980s and dealt with continuous LSIO problems. The papers [92]and [176] have recently reviewed the existing vast literature on the parametric modelassociated with the pseudometric d defined in (2.4) for LSIO, LIO, and CIO. Moredetails can be found in the remarks to Sects. 4–6.

2.2 Modeling Uncertain Portfolio Selection

Assume that C euros are to be invested in a portfolio comprised of n assets (shares,stocks, securities). Let ri be the return per 1 euro invested in asset i 2 f1; : : : ; ngduring a period of time (e.g., 1 month or 1 year). Since these returns are not knownin advance, r D .r1; : : : ; rn/ is an uncertain vector. The decision variables are theamount of euros to be invested in the i th asset, denoted by xi , i D 1; : : : ; n. Any


feasible portfolio x D .x1; : : : ; xn/ satisfies xi � 0; i D 1; : : : ; n,Pn

iD1 xi D C ,and other linear constraints, at least one of them being uncertain. In this section wediscuss portfolio models which are robust w.r.t. the constraints; one of these modelsis mixed whenever r is interpreted as a random (fuzzy, interval, parametric) vectorwhile it is a pure robust model when the task consists of maximizing the worstpossible return of the portfolio. Recalling (2.1), we consider an LSIO uncertainproblem of the form

P0 W maxx2Rn r

0x WDnX

iD1rixi s.t. a0

t x � bt ; t 2 T;

where the uncertainty exclusively affects the objective function. We assume thefeasibility of P0, i.e., the nonemptiness of the feasible set F � ˚

x 2 RnC W

PniD1

xi D C g.We will get a probabilistic model by interpreting each rate of return ri as a

random variable with expected value E Œri �, 1; : : : ; n. In the classical model ofstochastic optimization for the portfolio problem, whose history can be traced backto the 1950s, the decision maker solves

P1 W maxx2Rn E

r 0x WD

nX

iD1E Œri � xi s.t. a0

t x � bt ; t 2 T;

where the maximum is attained due to the compactness of F . This model is verysimple but absolutely unrealistic because it does not take risk into account. Now weassume that the investor wants to get a return less than � 2 I � R (I is an intervalcontaining any conceivable return) from a feasible portfolio x with a probabilityPr fr 0x � �g not greater than some p� 2 Œ0; 1�. The function � 7! p� captures theattitude of the decision maker toward risk and each inequality Pr fr 0x � �g � p�is called a Value-at-Risk (or a stochastic dominance) constraint. Thus we get thefollowing probabilistic problem:

P2 W maxx2Rn E Œr 0x�s.t. Pr fr 0x � �g � p�; � 2 I;

a0t x � bt ; t 2 T:

(2.6)

To the authors knowledge, the few available numerical methods for stochasticdominance constrained programs [137, 141] do not allow to solve the nonlinearsemi-infinite problem P2 except in particular cases. In fact, P2 is a hard problemeven in the simplest case that jI j D 1 and jT j < 1 (as in the classical chanceconstrained portfolio model), unless r has a multivariate normal distribution.

We shall get a robust optimization model by assuming that r ranges on a givenbounded set R � R

n. A pessimistic decision maker should maximize the worstpossible return of the portfolio x; i.e., the number infr2R r 0x. Thus, the robustcounterpart of P0 is the maxmin problem


P3 W supx2Rn

infr2R r

0x s.t. a0t x � bt ; t 2 T: (2.7)

Observing that infr2R r 0x is the greatest lower bound of the set fr 0x W r 2 Rg, thenonlinear semi-infinite program P3 turns out to be equivalent to the LSIO problem

P4 W sup.x;y/2RnC1 y

s.t. r 0x � y � 0; r 2 R;a0t x � bt ; t 2 T:

(2.8)

If � Dna0t x � bt ; t 2 T

ois continuous and R is a compact convex subset of

Rn, then P4 is a continuous LSIO problems, so that it can be solved by means of

the discretization methods mentioned in Sect. 1.3 except the interior point constraintgeneration algorithm (because the feasible set of P4 has an empty interior), althoughgrid discretization is not efficient whenever dimR � 3 (frequently, dimR D n).When T is finite it is preferable to use the interior point method of [215], exploitingthe affinity of the function r 7! r 0x � y for any couple .x; y/ 2 R

nC1. Numericalexperiments with P4, with T finite, are reported in the latter paper.

Next we propose an interval optimization model by assuming that the uncertain

set is the box R DYn

iD1 Œr i ; r i �, with ri < ri , i D 1; : : : ; n. We must determinethe range of the optimal value for all instances of the uncertain problem P0, i.e.,we have to solve a pessimistic and an optimistic counterparts of P0, but exploitingthe special structure of R. Since fri ; r ign is the set of extreme points of R, by aconvexity argument, the constraint subsystem fr 0x � y � 0; r 2 Rg is equivalent to

y �Xn

iD1 min frixi ; r ixig. So, introducing an auxiliary variable z 2 Rn, P4 can

be reformulated as

P5 W sup.x;y;z/2RnC1Cn y

s.t.nX

iD1zi � y � 0;

rixi � zi � 0; i D 1; : : : ; n;r ixi � zi � 0; i D 1; : : : ; n;a0t x � bt ; t 2 T:

The choice of a suitable LSIO numerical method to solve P5 is conditioned to the

properties of the systemna0t x � bt ; t 2 T

o. An optimistic decision maker expects

to get a return maxr2R r 0x from a feasible portfolio x. Thus she/he must solve thequadratic semi-infinite optimization problem

P6 W max.x;r/2R2n r 0xs.t. ri � ri � ri ; i D 1; : : : ; n;

a0t x � bt ; t 2 T;

whose difficulty lies in the lack of convexity of the objective function.


Finally, we construct a parametric model from a given nominal problem

P W maxx2Rn r

0x s.t. a0t x � bt ; t 2 T:

The space of parameters is ˘1 WD fag � fbg � Rn, that we can identify with R

n,equipped with the supremum distance d1. Obviously, F j˘1 W Rn � R

n is constant,while S j˘1 W Rn � R

n and # j˘1 W Rn ! R due to the compactness of F . We shallprove that S j˘1 is closed and usc at r while # j˘1 is continuous at r .

Let .rk/1kD1 � R

n and�xk�1kD1 � R

n be such that xk 2 S.rk/ for all k 2 N,rk ! r and xk ! x. Given x 2 F , we have

r 0kx

k � r 0kx for all k 2 N: (2.9)

Taking limits in (2.9) as k ! 1, we get r 0x � r 0x, so that x 2 S.r/. This showsthat S is closed at r . Moreover, S is equibounded at r in the sense that there existsU 2 Nr such that [fS.r/ W r 2 U g is bounded (take, e.g., U D R

n). Since anyequibounded set-valued mapping which is closed at a given parameter is usc at thatparameter, S turns out to be usc at r .

The following example shows that S j˘1 is not necessarily lsc at r W if F DŒ0; 1�2 and r D .0; 1/, then S.r/ D Œ0; 1� � f1g while S.r/ � f.0; 1/ ; .1; 1/g forr … cone frg sufficiently close to r , so that S shrinks abruptly close to r .

Concerning the optimal value function, since

# .r/ D maxx2F r

0x;

# coincides with the support function �F of the compact convex set F (i.e.,�F .r/ D supx2F x0r), # is a finite-valued positively homogeneous convex function,which implies # 2 C .Rn/. Thus, # is continuous at r .

Remark 2.2.1 (Antecedents). Examples of P2, in (2.6), with real data from USstock markets, with T finite, are discussed and solved in [68] and [69]. Alternativemodels are obtained by replacing the Value-at-Risk constraints in P3, in (2.7), byconvex approximations involving moments instead of probabilities. The tightestapproximations are provided by a risk measure introduced by Ben-Tal and Teboulle[17] and later popularized by Rockafellar and Uryasev [206] under the nameof conditional Value-at-Risk measure. This way P2 is replaced by a suitableconvex semi-infinite optimization (CSIO) problem which admits an equivalentLSIO reformulation under mild assumptions [71].

There exists a wide literature on fuzzy models for the portfolio problem (see,e.g., the recent surveys [225, 238]).

The returns r 0i s are interpreted as trapezoidal fuzzy numbers in [168,170], where

T is finite. In [224] the r 0i s are interpreted as LR-fuzzy numbers of different forms.

This type of fuzzy numbers were defined by Dubois and Prade [77] in an axiomaticway, enumerating the properties of two auxiliary functions, denoted by L and R,


which are required to be even, decreasing, usc on their supports, and take value 1at 0. The first model proposed in [224] consists of minimizing certain measure ofthe risk of achieving a return that is less than the return � of a given riskless asset(i.e.,

PniD1 xi ri � �) subject to a0

t x � bt , t 2 T . The second model replacesthis unique fuzzy constraint with infinitely many linear constraints giving rise to acertain LSIO problem. Numerical experiments with the latter model and T finitehave been reported in [224], taking real data from the Spanish stock market andmaking use of the simplex-like method in [169].

Remark 2.2.2 (An Open Problem in Uncertain LSIO). There is an absolute lack ofsoftware implementations and applications of post-optimality techniques to solvereal-world LSIO problems.

Chapter 3Robust Linear Semi-infinite Optimization

For many finite optimization problems, numerical methods can be compared fromthe complexity point of view, i.e., computing upper bounds on the number ofiterations, arithmetic operations, etc., necessary to get an optimal solution, or an"-optimal solution, in terms of the size of the problem. This methodology can hardlybe applied in LSIO because it is not evident how to define the size of the triplet.a; b; c/ representing the data of a problem like (1.1) despite the seminal results onthe complexity of the interior point constraint generation algorithm in [182, 192].On the other hand, the robust counterpart of an uncertain LSIO problem seldomenjoys the strong assumptions which are necessary to apply reduction or feasiblepoint methods. For this reason we identify, in this framework, tractability of a givenLSIO problem with satisfaction of the conditions guaranteeing the viability of thediscretization methods, namely:

• Continuity of the problem (main ingredient of any convergence proof).• The density assumption and low dimension of the index set (for grid discretiza-

tion algorithms).• The boundedness of the optimal set (for central cutting plane algorithms).• The full dimension and boundedness of the feasible set (for the interior point

constraint generation algorithm).

In this chapter we consider given an uncertain LSIO problem

P0 W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;

where the uncertain data may be either the constraints or the objective function, with

c D c 2 Rn (a fixed vector) in the first case and .a; b/ D

�a; b

�2 .Rn/T � R

T

(a fixed function) in the second one. The treatment of problems where all the dataare uncertain is a combination of both models. In each case, we associate with P0 aLSIO problem PR called robust (or pessimistic) counterpart whose correspondingcost vector, index set, feasible set, optimal solution set, first moment cone, and


39

40 3 Robust Linear Semi-infinite Optimization

characteristic cone are denoted by cR, TR, FR; SR, MR, and KR, respectively. TheHaar dual problem of PR is denoted byDR. We also associate with P0 a robust dualproblem (also called optimistic counterpart)DR such that the weak duality is alwayssatisfied while the strong duality holds under mild conditions. We also examine theconsistency of PR and its numerical tractability.

3.1 Uncertain Constraints

The LSIO problem P0 in the face of data uncertainty in the constraints can becaptured by a parameterized LSIO problem of the form

P u W infx2Rn c0xs.t. v0

t x � wt ;8t 2 T;

where c 2 Rn and u D .v;w/ W T ! R

n � R represents a selection of a givenuncertain set-valued mapping U W T � R

nC1 (in short, u 2 U ). Let Ut WD U.t/ �RnC1 for all t 2 T . Hence, in this robust model, the uncertainty set is the graph of

U , that is, gphU D f.t; ut / W ut 2 Ut ; t 2 T g.A robust decision maker facing uncertainty in the constraints intends to guarantee

the feasibility of her/his decisions, so that the robust counterpart of the parametricproblem .P u/u2U is the deterministic problem

PR W infx2Rn c0xs.t. v0

t x � wt ; ut D .vt ;wt / 2 Ut ; t 2 T; (3.1)

where the uncertain constraints are enforced for every possible value of the datawithin the prescribed uncertainty set gphU . Denoting TR WD S

t2T Ut , we can alsowrite

PR W infx2Rn c0xs.t. v0x � w; .v;w/ 2 TR:

(3.2)

So the feasible set, the characteristic cone and the first moment cones of PR areFR D fv0x � w; .v;w/ 2 TRg,

KR D cone fTR [ f.0n;�1/gg and MR D cone ProjRn .TR/ ; (3.3)

respectively.

Example 3.1.1. For illustration purposes, let us revisit Example 1.1.1 withc D .�1;�1/, where one considers perturbations, v1 .t/ and v2 .t/, of the nominalLHS coefficients of t 2 0; �

2

, � cos tand � sin t , such that jv1 .t/C cos t j � ˛

and jv2 .t/C sin t j � ˛, with 0 < ˛ < 1 (˛ can be interpreted as an upper bound

3.1 Uncertain Constraints 41

for the rounding errors caused by the subroutines providing approximate values ofthe LHS coefficients corresponding to indices t 2 0; �

2

). In terms of the robust

model considered in this section, the index set is T D 0; �2

[ f2; 3g, the uncertainset-valued mapping U W T � R

3 is

Ut D8<

:

Œ� cos t � ˛;� cos t C ˛� � Œ� sin t � ˛;� sin t C ˛� � f�1g ; t 2 0; �2

;

f.1; 0; 0/g ; t D 2;f.0; 1; 0/g ; t D 3;

and the robust counterpart of P0 is

PR W infx2Rn �x1 � x2s.t. v0

t x � wt ; .t; .vt ;wt // 2 gphU : (3.4)

If 0 < � < 1p2C2˛ and .t; .vt ;wt // 2 gphU , one has

hvt ; .�; �/i D � .v1 .t/Cv2 .t// � �� .cos tC sin tC2˛/ � ��p

2C2˛�> �1;

so that PR satisfies SCQ. Then, since gphU is compact, PR satisfies the FMCQ tooand so the characteristic cone KR of PR is closed.

A more realistic choice of U for P0 derives from the fact that the trigonometricfunctions are computationally approached via Taylor’s formula. So, we could takeU2 and U3 as above and

Ut D��1C t 2

2� t 4

24;�1C t 2

2

��t;�t C t 3

6

�� f�1g ; for all t 2

h0;�

2

i:

Now, for each fixed selection u D .v;w/ 2 U , the dual of P0 is the uncertainoptimization problem

Du W sup�2R.T /

C

(X

t2T�twt W

X

t2T�tvt D c

)

:

The optimistic counterpart of .Du/u2U is given by

DR W supuD.v;w/2U�2R.T /

C

(X

t2T�twt W

X

t2T�tvt D c

)

:

By construction,

v.DR/ � v.PR/: (3.5)


We say that robust duality holds for P0 whenever (3.5) holds with equality andDR is solvable, i.e.,

infx2FRhc; xi D max

uD.v;w/2U�2R.T /

C

(X

t2T�twt W

X

t2T�tvt D c

)

(3.6)

whenever the first member of (3.6) is finite. This is the “primal worst value” whilethe second member is the “dual best value,” so that robust duality means “primalworst equals dual best” with dual attainment.

We associate with PR the robust moment cone

MR WD[

uD.v;w/2Uconef.vt ;wt / ; t 2 T I .0n;�1/g:

Observe that MR is generally neither convex nor closed and it is related to KR bythe equation KR D convMR.

Now we are in a position to compare the optimistic counterpartDR with the Haardual DR of the pessimistic counterpart of P0 W

DR W sup�2R.gphU/

C

P.t;ut /2gphU �.t;ut /b.t;ut /

s.t.P

.t;ut /2gphU �.t;ut /a.t;ut / D c:Recall (see Sect. 1.2) that DR is equivalent to supy2R fy W .c; y/ 2 KRg in the sensethat both problems have the same optimal value and are simultaneously solvable ornot. By Theorem 1.2.1, v.PR/ D v.DR/ wheneverKR is closed. IfMR is closed andconvex, then KR D convMR DMR is closed too and so

v.PR/ D v.DR/

D max fy W .c; y/ 2 KRgD max

˚y W .c; y/ 2MR

�

D v.DR/:

We have thus proved the following robust duality theorem.

Theorem 3.1.1 (Global Robust Duality). Let FR ¤ ;. Then, robust duality holdsfor P0 whenever the robust moment cone MR is closed and convex.

The following result is a local version of the robust duality theorem.

Theorem 3.1.2 (Local Robust Duality). Suppose that the LFMCQ holds at x 2FR and Ut is convex for all t 2 T . Then, there exists a feasible solution

�.v;w/ ; �

�

for DR such that c0x DPt2T �twt .

In most real situations the uncertain set-valued mapping U takes the form

Ut WD�at ; bt

�C ˛tZ; t 2 T; (3.7)


where�a; b

�2 WD .Rn/T � R

T , ˛ 2 RTC and Z � R

nC1 is a compact

set such that 0nC1 2 Z. Denote by F the feasible set of the nominal systemna0t x � bt ; t 2 T

o. From now on in this section we assume that F ¤ ; (otherwise

PR is inconsistent for any ˛). The uncertainty with affine data perturbation occursin many real situations. It covers many commonly used data uncertainty sets suchas norm uncertainty set, box uncertainty set, and ellipsoidal uncertainty set wherethe set Z is a unit ball, a box, and an ellipsoid, respectively (see [14]). We mustguarantee the robust feasibility, i.e., the consistency of the robust counterpart PR,through conditions that can be expressed in terms of the data. Observe that both Uand PR depend on the parameter ˛ 2 R

TC.We first consider the uncertainty set-valued mapping U in (3.7) with ˛ 2 RC

(identified here with a constant mapping) and Z D B, where B denotes the closedunit ball for some norm k�k in R

nC1, that is,

Ut WD�at ; bt

�C ˛B; t 2 T: (3.8)

Observe that if PR is consistent for ˇ > 0, then PR is consistent also for � , for anynonnegative � < ˇ. Hence, the set f˛ 2 RC W PR is consistent for ˛g is an intervalwhose minimum element is 0.

The radius of consistency of the robust counterpart associated with U is

� .U/ WD sup f˛ 2 RC W PR is consistent for ˛g : (3.9)

The supremum in (3.9) cannot be C1 since, given t 2 T , .0n; 1/ 2�at ; bt

�C ˛B

for a positive large enough ˛, in which case the corresponding problem PR is notconsistent. Moreover, this supremum may not always be attained.

The next result provides a formula for the radius of consistency which involves

the so-called hypographical set [49] of the systemna0t x � bt ; t 2 T

o, defined as

H�a; b

�WD conv

n�at ; bt

�; t 2 T

oC RC f.0n;�1/g :

Since we are assuming that FR is consistent, F ¤ ;, so that, by the exis-tence theorem, 0nC1 cannot be an interior point of the characteristic cone ofna0t x � bt ; t 2 T

o, which obviously contains H

�a; b

�. So, denoting by d the dis-

tance associated with k�k, one has d�0nC1; bdH

�a; b

��D d

�0nC1; clH

�a; b

��.

Theorem 3.1.3 (Calculus of the Consistency Radius). Let U be as in (3.8). Thenthe equation

� .U/ D d�0nC1; clH.a; b/

�


holds under any of the following conditions:

(i) f.at ; bt /; t 2 T g is a bounded set without isolated points.(ii) B is the Euclidean unit ball and f.at ; bt /; t 2 T g is compact.

(iii) B is a polytope and f.at ; bt /; t 2 T g is finite.

The inequality � .U/ � d�0nC1; clH.a; b/

�follows from the identification of

the selections of U with perturbations of�a; b

�and the application of Theorem 6.2.2

(which is valid for any norm in RnC1). Recall that f.at ; bt /; t 2 T g is compact

whenever the nominal systemna0t x � bt ; t 2 T

ois continuous and that the unit ball

B is a polytope for the norms k�k1 and k�k1. Example 6.2.1 below illustrates the

computation of d2�0nC1; clH

�a; b

��for Example 1.1.1.

Theorem 3.1.4 (Attainability of the Consistency Radius). If 0CF is a linearsubspace, then the supremum in (3.9) is attained.

Sketch of the Proof. We assume w.l.o.g. that � .U/ > 0. Let .˛k/1kD1 � RCC be

such that ˛k ! � .U/. Then, for each k 2 N there exists xk 2 Rn such that

v0t xk � wt � 0 for all .vt ;wt / 2

�at ; bt

�C ˛kB; t 2 T:

This implies that

a0t xk � bt C inf

.ct ;dt /2B˛k�c0t xk � dt

� � 0 for all t 2 T:

Let �; ı 2 RCC be such that �B2 � B �ıB2. Then

inf.ct ;dt /2B

˛k�c0t xk � dt

� � inf.ct ;dt /2ıB2

˛k�c0t xk � dt

� � �ı˛k k.xk;�1/k2 :

So, one has

a0t xk � bt � ı˛k k.xk;�1/k2 � 0 for all t 2 T: (3.10)

Since .xk/1kD1 is bounded, we can assume w.l.o.g. that xk ! x. Taking limits in

(3.10) we get a0t x � bt � ı� .U/ k.x;�1/k2 � 0 for all t 2 T , so that x 2 FR. Thus

the supremum in (3.9) is attained (for the details, see [96, Proposition 3.7]).

Observe that, for the selection�a; b

�associated with the constraint system in

Example 1.1.1, one has 0CF D f02g, so that the supremum in (3.9) is attained.The next corollary shows that � .U/ > 0 for the model (3.7) whenever

˛ is a bounded function, and its proof is a straightforward application ofTheorem 3.1.3.


Corollary 3.1.1 (Consistency of the Robust Counterpart). Let U in (3.7) be such

that kzk < � for all z 2 Z and assume that�a; b

�and B satisfy one of the

conditions (i)–(iii) in Theorem 3.1.3. Then PR is consistent for any ˛ 2 RTC such

that

supt2Tj˛t j � ��1d

�0nC1; clH

�a; b

��:

Finally in this section, we consider several issues concerning the numericaltreatment of PR. Obviously, PR is continuous if and only if TR is compact, in whichcase PR can be solved via discretization. Concerning the relevant properties of thefeasible set FR, recalling the characterization of the properties of the feasible setof a linear system in terms of its characteristic cone and (3.3), we have that FR isa convex body if and only if cl coneTR is pointed and .0n;�1/ 2 int coneTR, inwhich case PR can be solved by means of the interior point constraint generationalgorithm. Similarly, concerning the optimal set, SR is bounded if and only ifc 2 int cone proj

Rn.TR/ by [102, Corollary 9.3.1], in which case PR can be solvedby some central cutting plane algorithm. The next result concerns the tractability ofthe robust counterpart under uncertainty with affine data perturbation.

Theorem 3.1.5 (Tractability of the Robust Counterpart). Let U be as in (3.7)

withna0t x � bt ; t 2 T

ocontinuous, ˛ 2 C.T /, and Z � R

nC1 compact. Then, the

following statements hold:

(i) PR is continuous provided it is consistent.(ii) If Z is a convex body and ˛ 2 R

TCC, then PR satisfies the density assumption.(iii) If Z D conv f.ri ; si / ; i 2 I g, where I is a finite set, then

FR Dnx 2 R

n W .at C ˛t ri /0 x � bt C ˛t si ; .t; i/ 2 T � Io: (3.11)

The assumption of (ii) guarantees that PR can be solved via grid discretiza-tion. The assumption of statement (iii) holds, e.g., whenever Z D B1, withI D f1; : : : ; 2ng. The next example shows that (3.11) may provide a reformulationof PR as LSIO problem with an index set of the same dimension as T .

Example 3.1.2. Consider the uncertain LSIO problem in Example 3.1.1, with anapproximation error not greater than a given ˛ 2 Œ0; 1�. Then, we have PR as in(3.4). Moreover, since B1 D conv f.˙1;˙1/g, Theorem 3.1.5(iii) allows us toreformulate PR as

P 1R W infx2R2 �x1 � x2

s.t. .˛ � cos t / x1 C .˛ � sin t / x1 � �1; t 2h0;�

2

i;

� .˛ C cos t / x1 C .˛ � sin t / x1 � �1; t 2h0;�

2

i;


Fig. 3.1 Generators of MR

.˛ � cos t / x1 � .˛ C sin t / x1 � �1; t 2h0;�

2

i;

� .˛ C cos t / x1 � .˛ C sin t / x1 � �1; t 2h0;�

2

i;

x1 � 0 .t D 2/; x2 � 0 .t D 3/; (3.12)

where we can replace the identical index intervals in (3.12) by T1 D0; �

2

,

T2 D2�; 5�

2

, T3 D

4�; 9�

2

, and T4 D

6�; 13�

2

, consecutively. This way

we get an equivalent continuous LSIO problem P 2R whose feasible set F 2

R is fulldimensional and compact (as P 2

R satisfies SCQ and F 2R � F ), its solution set

S2R is compact too, and its index set�S4

iD1 Ti�[ f2; 3g satisfies the density

assumption and is contained in the one-dimensional interval Œ0; 21�. Figures 3.1and 3.2 represent the set of LHS vector coefficients ofPR andP 2

R (i.e., the generatorsof the respective first moment conesMR andM2

R), for ˛ D 0:2 (the curves in Fig. 3.2are labeled as the corresponding index intervals). Thus, P 2

R could be solved viagrid discretization, central cutting plane methods, and the interior point constraintgeneration algorithm.

Remark 3.1.1 (Antecedents and Extensions). The tractability of the robustcounterparts of ordinary optimization problems has been analyzed in [14–16],etc., while robust duality theorems for LO and ordinary convex optimization havebeen given in [12] and [150], among others. The proofs of Theorems 3.1.1 and 3.1.2can be found in [95, Theorem 1] and [96, Corollary 2.7]. Theorem 3.1.3(ii) is [96,Theorem 3.3], while statements (i) and (iii) can be derived from the arguments in[96, Remark 3.4]. Theorem 3.1.4 generalizes [96, Proposition 3.7] to arbitrary normswhile Theorem 3.1.5 is an adaptation of [96, Proposition 4.2]. In [95, Proposition 1]it is shown that MR is convex in the case of affinely parameterized data uncertainty

3.2 Uncertain Objective 47

Fig. 3.2 Generators of M2R

(defining this setting as in [14]) while [95, Proposition 2] shows that MR is closedunder a robust SCQ together with suitable topological requirements on the indexset and the uncertainty set of the problem.

Duality theorems for robust multiobjective LSIO under uncertain constraints canbe found in [96], and for robust convex optimization problems posed in locallyconvex spaces in [27]. The latter paper includes an application to best approximationproblems with constraint data uncertainty. An extension of Theorem 3.1.5 to robustmultiobjective LSIO can be found in [96, Proposition 15].

3.2 Uncertain Objective

We denote by F and M the feasible set and the first moment cone ofna0t x � bt ; t 2 T

oand by �C the support function of C . We assume that F ¤ ;.

Then the robust counterpart of P0 is the CSIO problem

infx2Rn �C .x/ D supc2C c0xs.t. a0

t x � bt ; t 2 T;or, equivalently, the LSIO problem

PR W inf.x;y/2RnC1 y

s.t. �c0x C y � 0; c 2 C;a0t x � bt ; t 2 T; (3.13)


so that cR D .0n; 1/, FR D epi �C \�F � R

�, TR D T [C (we can assume w.l.o.g.

that C \ T D ;),MR D cone f.�C/ � f1g/ [ f.at ; 0/ ; t 2 T ggD �M � f0g�C cone f.�C/ � f1gg ;

and

KR D coneh..�C/ � f1g � f0g/ [

n�at ; 0; bt

�W t 2 T

o[ f.0n; 0;�1/g

i:

Obviously, PR is consistent if and only if F \ dom �C ¤ ;. In particular, PR

is consistent whenever C is bounded. The robust duality theory is a straightforwardspecification of the one exposed in Sect. 3.1, just defining the uncertainty set-valuedmapping U W T [ ft0g � R

nC1, with t0 … T , such that Ut0 D .�C/ � f1g � f0gand Ut D

n�at ; 0; bt

�ofor all t 2 T . Concerning the suitable numerical treatment

of PR, it depends on its relevant properties: continuity of the constraint system,density assumption, boundedness of the optimal set SR, and boundedness and fulldimensionality of FR.

Typically, T � Rm for somem � n andC D cC˛Z, where ˛ 2 RCC andZ is a

given bounded set of Rn. For instance, ifZ is the closed unit ball for some norm k�kin R

n, then C is a ball centered at c, �C is finite-valued and �C .x/ D c0xC˛ kxk�,where k�k� denotes the dual norm (recall that k�k2 coincides with its dual norm whilek�k1 and k�k1 are dual to each other). Alternatively, if Z is a cartesian productof (possibly degenerate) closed intervals containing 0, C is a box not necessarilysymmetric w.r.t. c. In either case, we can assume w.l.o.g. that T � R

n (otherwise,we can replace T with T � f0n�mg) and C \ T D ;. If the nominal problem P

in (2.2) is continuous, then PR is continuous too. If, additionally, P satisfies thedensity assumption (e.g., T is convex), then TR satisfies the density assumption too.Concerning the dimension of TR, it is the maximum of the dimensions of T and C .If x 2 F , then .x;maxc2C c0x/ 2 FR, so that PR is consistent (with unboundedfeasible set FR) if and only if P is consistent. Finally, concerning SR, it is boundedif and only if

.0n; 1/ 2 int�M � f0g�C cone f.�C/ � f1gg : (3.14)

Example 3.2.1. Consider the same problem as in Example 3.1.2, but assuming nowthat the source of uncertainty is the objective function. Take C D .�1;�1/C ˛B2,with ˛ 2 Œ0; 1�. Then the robust counterpart of P0 is

PR W inf.x;y/2R3 ys.t. .1 � ˛u1/ x1 C .1 � ˛u2/ x2 C y � 0; u 2 B2;

� .cos t / x1 � .sin t / x2 � �1; .t; v/ 20; �

2

� Œ2; 3� ;x1 � 0 .t D .2; 0//; x2 � 0 .t D .3; 0//:

3.2 Uncertain Objective 49

Obviously, PR is continuous, the index set B2 [0; �

2

� Œ2; 3� [ f.2; 0/ ; .3; 0/gsatisfies the density assumption and is contained in the two-dimensional intervalŒ0; 3�2, and its feasible setFR is an unbounded closed set. Moreover, sinceM�f0g DR2 � f0g and .1; 1; 1/ 2 .�C/� f1g, �M � f0g�C cone f.�C/ � f1gg D R

2 �RC.Thus (3.14) holds and the optimal set SR is bounded. Consequently, PR could besolved by means of grid discretization or central cutting plane algorithms.

Remark 3.2.1 (Antecedents and Extensions). The reader is referred to theantecedents mentioned in Remark 3.1.1. In particular, [95, Corollary 2] establishesa robust duality theorem for robust LSIO with uncertain objective and constraints.Robust multiobjective LO with uncertain objective has been analyzed in [97].

Remark 3.2.2 (Two Open Problems in Robust LSIO).

1. Complexity analysis of the robust counterparts of particular types of LSIOproblems.

2. Systematic analysis of the tractability of the robust counterparts of particulartypes of LSIO problems.

Chapter 4Sensitivity Analysis

From now on we assume that a consistent nominal LSIO problem P as in (2.2) isgiven, with dual problem

D W sup�2R.T /X

t2T �tbt

s.t.X

t2T �tat D c;�t � 0; t 2 T:

Denote by � D .c; a; b/ the parameter representing P in the space of parameters˘ D R

n � .Rn/T � RT , by M.�/ D cone fat ; t 2 T g the first moment cone of P ,

and by K.�/ D conen�at ; bt

�; t 2 T I .0n;�1/

othe characteristic cone of P . We

associate with each triplet � D .c; a; b/ 2 ˘ representing a perturbation of � theprimal LSIO problem

P .�/ W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;and its dual one

D .�/ W sup�2R.T /X

t2T �tbt

s.t.X

t2T �tat D c;�t � 0; t 2 T:

Recall that in the parametric setting we denote by F .�/, S .�/, and # .�/

the feasible and the optimal sets of P .�/, and its optimal value, respectively.Analogously, we represent by FD .�/, SD .�/, and #D .�/ the correspondingobjects of the dual problem D .�/.

The objective of this chapter consists of determining affinity regions of therestriction of the optimal value functions # (or #D) to some subspace of ˘ , andtheir estimations under three different types of perturbations of the nominal data� D .c; a; b/.


51

52 4 Sensitivity Analysis

4.1 Perturbing the Objective Function

Suppose that only the objective function of P can be perturbed. Then the space

of parameters is formed by the triplets of the form � D�c; a; b

�, with parameter

c 2 Rn, i.e., we can identify the subspace of parameters with R

n.The perturbed problems of P and D to be considered in this section are

P .c/ W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;

and

D .c/ W sup�2R.T /

C

P

t2T�tbt

s.t.P

t"T

�tat D c;

with optimal sets S .c/ and SD .c/, and optimal values # .c/ and #D .c/, respec-tively (obviously, P .c/ is continuous when P is continuous). With this notation,the effective domain of the dual optimal value function #D is the first momentcone, M .�/, and the optimal values of the nominal problem P and its dual Dare # .c/ and #D .c/, respectively. We denote by @# W Rn � R

n the set-valuedmapping which assigns to each c 2 R

n the concave subdifferential of the optimalvalue function of # at c.

The next result provides an estimation of the increment of # .c/ D v�P�

whenonly the objective function of P is perturbed.

Theorem 4.1.1 (Estimation of the Optimal Value). Let P be a consistent LSIOproblem with characteristic cone K .�/. Then the following statements hold:

(i) hypo#D D K .�/.(ii) hypo# D clK .�/.

(iii) rintM .�/ � dom# � clM .�/.(iv) @# D S .(v) c 2 intM .�/ if and only if S .c/ is bounded.

From (i) and (ii) we conclude that the concave function # is the usc hullof #D . From (iii), (iv), and [205, Theorem 23.4], we get an expression for thedirectional derivative of # at a c 2 rintM .�/ D rint dom# , where dom# Dfc 2 R

n W # .c/ > �1g W given a direction d 2 Rn, one has

# 0 .cI d/ WD lim�#0

# .c C �d/ � # .c/�

D � lim�#0�# .c C �d/C # .c/

�

D � .�#/0 .cI d/ D � supu2@.�#/.c/

d 0u D infx2S.c/

d 0x: (4.1)

4.1 Perturbing the Objective Function 53

In particular, by (v), # 0 .cI d/ D minx2S.c/ d 0x whenever c 2 intM .�/. Applying(4.1) to P such that c 2 rintM .�/, we conclude that a suitable estimation of # .c/is given by

# .c/ ' # .c/C infx2S.c/

hc � c; xi ; (4.2)

for c in the proximity of c. When, c 2 intM .�/ and P has a unique optimalsolution x, from (iv) and (4.2) # is differentiable at c, with r# .c/ D fxg and# .c/ ' # .c/C hc � c; xi.

The next three results extend to LSIO classical results on sensitivity analysis inLO, where exact formulas have been given for # .c/ in the proximity of the nominalparameter c. The extension of these results to LSIO requires the introduction ofsuitable partition concepts.

Given a convex or concave positive homogeneous extended function f , wedefine the affinity cone of f at z 2 .dom f /Ÿ f0ng, denoted by Lz, as the largestrelatively open convex cone containing z on which f is affine (i.e., simultaneouslyconvex and concave). An argument based on convex analysis tools shows that thecollection

L .f / WD fLz W z 2 .dom f /Ÿ f0ngg

of affinity cones of a convex positive homogeneous function f WRn ! R constitutesa partition of .dom f /Ÿ f0ng. So, we call L .f / the affinity partition of f . Sincethe above statement remains true for concave positive homogeneous functions and# is a superlinear (i.e., a usc concave positive homogeneous) function, we get thefollowing result:

Theorem 4.1.2 (Affinity Partition of the Optimal Value Function). The classof relatively open cones L .#/ constitutes a partition of .dom#/Ÿ f0ng in regionswhere # is affine.

Thus, Lc 2 L .#/ is the largest relatively open convex set (actually a convexcone) on which # is affine. This information is more precise than the one given by(4.2), but it only applies on a neighborhood of c whenever Lc is full dimensional.This situation is approached in the following theorem.

Theorem 4.1.3 (Affinity on Neighborhoods). # is affine on some neighborhoodof c if and only if P has a strongly unique solution. In such a case, if S .c/ D fxg,then # .c/ D hx; ci for all c 2 Lc (an open convex cone containing c).

Combining Theorem 4.1.3 with the equation

v�P� D sup

y2Rfy W .c; y/ 2 clK .�/g

we get the following geometric characterization of the existence of strongly uniquesolution.


Corollary 4.1.1. P has a strongly unique solution if and only if the point

�c; supy2R fy W .c; y/ 2 clK .�/g� 2 R

nC1

belongs to the relative interior of a facet of clK .�/.

Example 4.1.1. Consider the parametric problem

P .c/ W infx2R2 c0xs.t. � .cos t / x1 � .sin t / x2 � �1; t 2

0; �

2

;

x1 � 0 .t D 2/; x2 � 0 .t D 3/:

The affinity partition L .#/ of # is formed by the images by ProjR2 of the exposed

faces of hypo# D clK .�/, eliminating the origin (see Fig. 4.1), so that

dimLc D�1; if c 2 R2� [ .RC � f0g/ [ .f0g � RC/

Ÿf02g;

2; otherwise.

The linearity cones are as follows for the three different objective functionsconsidered in Example 1.1.1:

Case (a) c D .1; 1/ W # 0 on the open convex cone Lc D R2CC. Since # is affine

on a neighborhood Lc of c; P .c/ has a strongly unique optimal solution(the origin 02).

Case (b) c D .�1;�1/ W # .c/ D c1Cc2p2

for all c 2 Lc D RCC.�1;�1/.Case (c) c D .1; 0/ W # .c/ D 0 for all c 2 Lc D RCC .1; 0/.

Table 4.1 shows the close relationship between affinity cones of the points c 2R2Ÿ f02g, and the maximal optimal partitions and the optimal sets of P .c/. More

precisely, all problems P .c/ corresponding to vectors c from a given affinity coneL 2 L .#/ have the same maximal optimal partition. This cannot be a general ruleas LSIO problems may have or not maximal optimal partition.

From the definition of L .#/, it is obvious that if˚ci ; i 2 I� � L 2 L .#/ implies

that # is affine on conv˚ci ; i 2 I�. The next result is a counterpart of this statement

involving optimal partitions (in Example 4.1.1, the problems associated with ci , i 2I , have the same maximal optimal partition if and only if

˚ci ; i 2 I� is contained in

some element of the affinity partition).

Theorem 4.1.4 (Affinity on Polytopes). Let˚ci ; i 2 I� � R

n, with I finite,be such that there exists a common optimal partition for the family of problems˚P�ci�; i 2 I� (e.g., they have the same maximal optimal partition). Then # is

affine on conv˚ci ; i 2 I�.

4.1 Perturbing the Objective Function 55

Fig. 4.1 Linearity cones partition

Table 4.1 Affinity partition in Example 4.1.1

Affinity cone Maximal optimal partition Optimal set

R2CC

�0; �

2

; f2; 3g ;;� f02g

f0g � RCC

�0; �

2

[ f2g ; f3g ;;� Œ0; 1�� f0gR�� RCC

�0; �

2

[ f2g ; f0; 3g ;;� f.1; 0/gR�� f0g �

0; �2

[ f2g ; f0g ; f3g� f.1; 0/gRCCc, c 2 R

2��

.TŸ f˛g ; f˛g ;;/, ˛ D arctan�c2c1

� n�� c1

kck2;� c2

kck2

�o

f0g � R��

�0; �

2

[ f3g ; ˚ �2

�; f2g� f.0; 1/g

RCC � R��

�0; �

2

[ f3g ; ˚ �2; 2�;;� f.0; 1/g

RCC � f0g �0; �

2

[ f3g ; f2g ;;� f0g � Œ0; 1�

The duality approach to sensitivity analysis provides the next result on affinityalong segments under assumptions implying the existence of optimal partition.

Theorem 4.1.5 (Affinity Along Segments). # is affine on a segment emanatingfrom c in the direction of d 2 R

nŸ f0ng if P and D are solvable, with v�P� D

v�D�, and the problem

Dd W sup�2R.T /

C;�2R

P

t2T�tbt C �v

�P�

s.t.P

t"T

�tat C �c D d;

is also solvable and v�P d

� D v�Dd

�.


By Theorem 1.2.1, the assumptions on Dd hold whenever its correspondingprimal problem

P d W infx2Rn d 0xs.t. a0

t x � bt ; t 2 T;c0x D v

�P�;

satisfies the FMCQ.

Example 4.1.2. Let P be the problem P .�1;�1/ of Example 4.1.1(b) and d 2 R2

such that 0 < kdk2 < 1. Here

P d W infx2Rn d 0xs.t. � .cos t / x1 � .sin t / x2 � �1; t 2

0; �

2

;

x1 � 0; x2 � 0; x1 C x2 Dp2:

Denoting by K .�/ and Kd the characteristic cones of P and P d , we have Kd DK .�/ C span

n�1; 1;p2�o

, and this cone is not closed (its closure is the half-

space x3 � x1 C x2). So, the reader may verify that neither Theorem 4.1.3 norTheorem 4.1.5 can be applied.

Observe that the maximal optimal partitions of c D .�1;�1/ and c C d D.�1C d1;�1C d2/, are

�TŸ

˚�4

�;˚�4

�;;� and .TŸ f˛g ; f˛g ;;/, which coin-

cide if and only if ˛ D arctan��1Cd2�1Cd1

�D �

4, i.e., d 2 RCCc. So, Theorem 4.1.4

applies whenever d 2 RCCc to conclude that # is affine on the segment Œc; c C d�.Thus, # is affine on the half-line RCCc (see Fig. 4.1).

Remark 4.1.1 (Antecedents and Sources). Theorem 4.1.1 on the properties of theoptimal value function has the antecedent of the duality theorem for superconsistentLSIO problems in [209], whose proof is quite schematic; a more detailed proofcan be found in [102, Theorem 8.1]. Conditions for the differentiability of # atc can be obtained from the identity # .c/ D ��F .�c/ and the results in [232,233]. In particular, if F is a compact convex set with nonempty interior, then #is differentiable except at the origin if and only if F is strictly convex (i.e., bdFcontains no segment).

Theorems 4.1.2 and 4.1.3 have no antecedent in LO; they have been proved in[110, Proposition 2.2] and [93, Theorem 1], respectively. Finally, Theorem 4.1.5extends a similar result of Gauvin [87] on sensitivity analysis in LO and wasproved in [93, Theorem 2], while Theorem 4.1.4 was shown in [110, Proposition4.1] and can be seen as an extension, to LSIO, of a large stream of papers onsensitivity analysis in LO from an optimal partition perspective [1,19,85,88,89,119–122,146–148,188,208]. Analogous extensions have been proposed for semidefiniteoptimization [117], conic optimization [230], and quadratic optimization [89].

4.2 Perturbing the RHS 57

4.2 Perturbing the RHS

Now we consider the parametric problems

P .b/ W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;

and

D .b/ W sup�2R.T /

C

P

t2T�tbt

s.t.P

t"T

�tat D c;

with respective optimal values # .b/ and #D .b/. Obviously, the optimal values of

the nominal problem P and its dual D are v�P� D #

�b�

and v�D� D #D

�b�

,

respectively, as P .b/ � D .c; a; b/ and P D P .b/ � D .c; a; b/. Noticethat the first moment cone of P .b/ coincides with the first moment cone M .�/

of P . So, if c 2 rintM .�/ (e.g., the feasible set F of P is bounded) and P .b/is consistent, then #D .b/ D # .b/ by Theorem 1.2.1, i.e., #D coincides with # ondomF .

Concerning the perturbations b W T ! R, we consider the space of parameters,identified with R

T , equipped with the pseudometric of the uniform convergence:

d1 . ; '/ WD supt2Tj .t/ � ' .t/j ; ; ' 2 R

T :

The next result can be seen as a RHS counterpart of Theorem 4.1.3.

Theorem 4.2.1 (Affinity on Neighborhoods). If # is linear on a certain neigh-borhood of b, then D has at most one optimal solution. Conversely, if there exist

x� 2 F�b�

, � > 0 and � > 0 such that

(i) c 2 A .x�/,(ii) fat ; t 2 T .x�/g is a basis of Rn, and

(iii) a0t x � bt C � for all x 2 x� C �B2 and t … T .x�/,

then # is the linear function # .b/ D c0x .b/ in a neighborhood of b, where x .b/ isthe unique solution of the system

˚a0t x D bt ; t 2 T .x�/

�.

The following result guarantees the affinity of # along segments under strongassumptions.

Theorem 4.2.2 (Affinity Along Segments). # is affine on a segment emanatingfrom b in the direction of a bounded function f 2 R

TŸ f0T g if P and D aresolvable with the same optimal value, the problem


Pf W inf.x;y/2RnC1 c0x C #�b�y

s.t. a0t x C bty � ft ; t 2 T;

is also solvable and has zero duality gap with Df , and there exists x� 2S�b�

such

that either T .x�/ D T or there exist two scalars � and � such that 0 < � �a0t x

� � bt � � for all t … T .x�/.

Theorem 4.2.3 (Affinity on Polytopes). Let conv˚bi ; i 2 I�, with I finite, be

such that there exists a common optimal partition for all the problems P�bi�, i 2 I .

Then # .b/ D #D .b/ is affine on conv˚bi ; i 2 I�.

Hence, if d 2 RT and there exists " > 0 such that P .b C "d/ has the same

optimal partition as P , then # .b/ D #D .b/ is affine on Œb; b C "d �.Example 4.2.1. We revisit Example 4.1.1(b), where T D 0; �

2

[f2; 3g. The primalproblem associated with b 2 R

T is

P .b/ W infx2R2 �x1 � x2s.t. � .cos t / x1 � .sin t / x2 � bt ; t 2

0; �

2

;

x1 � b2; x2 � b3;

and the nominal parameter is b 2 RT such that bt D �1 for all t 2 0; �

2

, and

b2 D b3 D 0. Recall that x D�

1p2; 1p

2

�and

�TŸ

˚�4

�;˚�4

�;;� are the unique

optimal solution and the maximal optimal partition of P�b�

, respectively.

Now we consider individual perturbations of the RHS of the constraints whichpreserve the optimality of x.

The constraint corresponding to t 2 0; �

2

is redundant in P

�b�

, so that

F�b C ˛t

�D F

�b�

for any ˛ � 0, where t denotes the characteris-

tic function of t . Even more, x 2 F�b C ˛t

�� F

�b�

for any ˛ �a0t x. Thus, S

�b C ˛t

�D fxg for any ˛ � a0

t x. Repeating the argument of

Example 1.2.1 one concludes that�TŸ

˚�4

�;˚�4

�;;� is the maximal optimal

partition of P�b C ˛t

�� D

�b C ˛t

�for any ˛ � a0

t x D � cos tCsin tp2

D� cos

�t � �

4

�.

The constraint corresponding to t D 2; 3 are not redundant, but it is still

true that�TŸ

˚�4

�;˚�4

�;;� is the maximal optimal partition of P

�b C ˛t

��

D�b C ˛t

�for any ˛ � a0

t x D � 1p2, t D 2; 3.

Then, by Theorem 4.2.3, # is affine on

conv

[

t2T

nb C ˛t W ˛ � a0

t xo!

:

4.3 Perturbing the Objective Function and the RHS 59

Remark 4.2.1 (Antecedents and Sources). Antecedents for the results in this sectionare the works on sensitivity analysis in LO from an optimal partition perspectivementioned in Remark 4.1.1. Theorems 4.2.1, 4.2.2, and 4.2.3 are [93, Theorem 4],[93, Theorem 5], and [110, Proposition 5.1], respectively.

4.3 Perturbing the Objective Function and the RHS

We associate with the given nominal problem P the space of parameters RT � Rn.

The primal and dual problems associated with a perturbation � D .a; b; c/ of thenominal parameter � D .c; a; b/ are

P .c; b/ W infx2Rn c0xs.t. a0

t x � bt ; t 2 T;

and

D .c; b/ W sup�2R.T /X

t2T �tbts.t.

X

t2T �tat D c;�t � 0; t 2 T;

with optimal values # .c; b/ and #D .c; b/, respectively. In order to describe thebehavior of # and #D , we define a class of functions after giving a brief motivation.

Let L be a linear space and let ' W L2 ! R be a bilinear form on L. LetC D conv fvi ; i 2 I g � L and let qij WD '

�vi ; vj

�, .i; j / 2 I 2. Then any v 2 C

can be expressed as

v DX

i2I�ivi ;

X

i2I�i D 1; � 2 R

.I /C : (4.3)

Then we have

' .v; v/ DX

i;j2I�i�j qij : (4.4)

Accordingly, given q W C � C ! R, where C D conv fvi ; i 2 I g � L, we saythat q is quadratic on C if 9qij 2 R, i; j 2 I , such that (4.4) holds for all v 2 Csatisfying (4.3).

Theorem 4.3.1 (Quadratic Behavior on Polytopes). Let˚�ci ; bi

�; i 2 I� �

Rn � R

T , with I finite, be such that there exists a common optimal partition forthe family of problems P

�ci ; bi

�, i 2 I . Then P .c; b/ and D .c; b/ are solvable,

# .c; b/ D #D .c; b/ on conv˚ci W i 2 I�� conv

˚bi W i 2 I� and # is quadratic on


conv˚�ci ; bi

� W i 2 I�. Moreover, if .c; b/ 2 conv˚ci W i 2 I� � conv

˚bi W i 2 I�,

then # .�; b/ and # .c; �/ are affine on conv˚ci W i 2 I� and conv

˚bi W i 2 I�,

respectively.

So, given .d; f / 2 Rn � R

T , if there exists " > 0 such that the prob-

lem P��c; b

�C " .d; f /

�has the same maximal optimal partition as P , then

# .c; b/ D #D .c; b/ is quadratic on the intervalh�c; b

�;�c; b

�C " .d; f /

i.

Moreover, #�c; b

�(# .c; b/) is an affine function of c on Œc; c C "d � (of b on

hb; b C "f

i, respectively).

Example 4.3.1. Let us consider simultaneous perturbations of the objectivefunction and RHS coefficients of the nominal problem of Examples 4.1.2and 4.2.1. Combining the arguments there, we get that # is quadratic onconv

˚�ci ; bi

� W i 2 I�, where˚ci W i 2 I� D RCCc and

˚bi W i 2 I� D conv

[

t2T

nb C ˛t W ˛ � a0

t xo!

:

Remark 4.3.1 (Antecedents and Sources). Once again the antecedents for theresults in this section are the works on sensitivity analysis in LO from anoptimal partition perspective mentioned in Remark 4.1.1. Theorem 4.3.1 is[110, Proposition 6.1]. Concerning the estimations of the optimal value throughdirectional derivatives, [234] deals with the continuous linear parametric problemP.�/ D �.�/ D .c; a.�/; b.�// (i.e., a.�/ and b.�/ are continuous functions of ton T which depend on a parameter �), the nominal problem being P D P.�/ D�.�/. The approach of the authors extends to LSIO some results in [118]. Underthe assumption of superconsistency of the nominal dual pair P.�/ � D.�/ (i.e.,SCQ and c 2 intM.�/), and other technical assumptions, Theorem 6 in [234]provides expressions for the right-hand and left-hand side derivatives of the optimalvalue function # at �. Concerning extensions, Shapiro [210, Theorem 3.2] providessufficient conditions for the right-hand side differentiability of the optimal valuefunction for differentiable CSIO problems and gives an explicit formula [210, (3.7)]for this derivative.

Remark 4.3.2 (Some Open Problems in Sensitivity Analysis of LSIO Problems).

1. Sensitivity analysis of LSIO problems under perturbations of a.2. Sensitivity analysis of LSIO problems under perturbations of the triple .c; a; b/.3. Numerical methods for the computation of affinity regions under perturbations

of c, b, and the couple .c; b/.

Chapter 5Qualitative Stability Analysis

5.1 Irrestricted Stability

In this section we analyze the continuity properties of F , S W ˘ � Rn and # W

˘ ! R at a given nominal problem � D .c; a; b/ under arbitrary perturbations ofall the data. The first contributions to the stability of continuous LSIO and generalLSIO problems were published in the 1980s (e.g., [29, 82]) and 1990s [103, 104],respectively, and the main results in these works are gathered in the monograph[102].

We have defined in Sect. 2.1.5 the three main concepts involved in the qualitativestability analysis of a set-valued mapping M W Y � X at y 2 domM(equivalently, M .y/ ¤ ;): lower and upper semicontinuity, and closedness.We introduce here some related concepts.

• Again when Y and X are pseudometric spaces, we can define the inner limit

lim infy!y M .y/ WD8<

:

x 2 X W 8.yr /1rD1 ! y an associated r0 existssuch that .yr /1rDr0 � domM;

and 9xr 2M .yr / 8r � r0 such that xr ! x

9=

;;

and the outer limit

lim supy!y M .y/ WD�x 2 X W 9.yr /1rD1 ! y and .xr /1rD1; xr 2M .yr / ;

such that xr ! x

�:

Following [207], we say that M is outer semicontinuous (osc) at y whenM .y/ � lim supy!y M .y/. Similarly, M is inner semicontinuous (isc) at y ifM .y/ � lim infy!y M .y/. Finally, we say that M is continuous at y 2 domMif M is isc and osc at y, i.e.,

lim infy!y M .y/ D lim supy!y M .y/ DM .y/ ;


61

62 5 Qualitative Stability Analysis

in which case we write

limy!y

M.y/ DM .y/ :

Inner semicontinuity of M at y is equivalent to the lower semicontinuity of M aty (in the sense of Berge), whereas the outer semicontinuity of M at y is equivalentto the closedness of M at y.

• We also say that M is inner semicontinuous at .y; x/ 2 gphM if for every openset U 3 x there exists V 2 Ny such that M.y/ \ U ¤ ; for all y 2 V , thatis, x 2 lim infy!y M .y/. Clearly, M is inner semicontinuous at y if and only ifM is inner semicontinuous at .y; x/ for every x 2M.y/.

We have already mentioned that F is closed at any � 2 domF , i.e., gphF isclosed. The fundamental result relative to F characterizes the lower semicontinuityof F at � in a variety of ways, which involve the previous concepts as well as thefollowing one:

• F is stable in Tuy’s sense at � if 0T … bd˚G .Rn/ � R

TC�, where G .x/ .:/ WD

s.x; :/; x 2 Rn (here s.x; :/ is the slack function at x, and R

T is assumed to beequipped with the topology of the uniform convergence on T ).

Theorem 5.1.1 (Lower Semicontinuity of the Feasible Set). Let � 2 domF . Thefollowing statements are equivalent:

(i) F is lsc at � I(ii) � 2 int domF I

(iii) sufficiently small perturbations of the RHS of � preserve its feasibility;(iv) � satisfies the SSCQ;

(v) 0nC1 … clC.�/, where C.�/ D convn.at ; bt /; t 2 T

o;

(vi) there exists V 2 N� such that dimF .�/ D dimF .�/ for all � 2 V

(dimensional stability)I(vii) there exists V 2 N� such that affF .�/ D affF .�/ for all � 2 V (affine

hull immobility)I(viii) � is stable in Tuy’s sense at � I

(ix) F is continuous at � .Moreover, in the case that 0n … bd conv fat ; t 2 T g, the following conditionis added to the list:

(x) there exists V 2 N� such that F .�/ is homeomorphic to F .�/ for all � 2 V(topological stability).

In [207] it is said that “upper semicontinuity differs from our outer semicontinu-ity and is seriously troublesome in its narrowness.” This is why the characterizationof the upper semicontinuity of F at � is a hard problem, conceptually solved in [45]developing some ideas in [104].

5.1 Irrestricted Stability 63

We have already stated that upper semicontinuity hardly holds when the imageis non-compact. In fact it requires that the perturbed feasible sets differ from thenominal one in a uniformly bounded manner, as Theorem 5.1.2 below shows.A preliminary step is the following lemma:

Lemma 5.1.1 (Uniform Boundedness of the Feasible Set). Let � 2 domF . IfF.�/ is bounded, F is uniformly bounded in some neighborhood of � .

Sketch of the Proof. Otherwise, there will exist a sequence .�r/1rD1 convergingto � and xr 2 F.�r/ such that kxrk2 � r , r D 1; 2; : : :. Then, if d is anaccumulation point of .xr= kxrk2/1rD1, it can easily be proved that d 2 0CF.�/and this contradicts the assumed boundedness of F.�/:

The following theorem is Theorem 3.1 in [104].

Theorem 5.1.2 (Upper Semicontinuity of the Feasible Set). F is usc at � 2domF if and only if there exist � > 0 and a neighborhood V 2 N� such that

F.�/Ÿ�B2�F.�/Ÿ�B2 for all � 2 V . (5.1)

Example 5.1.1. Let us consider the parameter � , in R2, whose constraint system is

� D ftx1 C x2 � � jt j ; t 2 Rg:Obviously, F.�/ D Œ�1; 1� � RC and

clK.�/ D conef.�1; 0;�1/; .1; 0;�1/; .0; 1; 0/g: (5.2)

If d1.�; �/ < C1 one can easily prove that x1 2 Œ�1; 1� for all x 2 F.�/, and ifd1.�; �/ < 1 and

� WD 2d1.�; �/1 � d1.�; �/

;

then x2 � �� for all x 2 F.�/ (take t D 0). Therefore, F.�/Ÿ.�C 1/B2�F.�/Ÿ.�C 1/B2 for all � such that d1.�; �/ < 1, and Theorem 5.1.2 appliesto conclude that F is usc at � .

Unfortunately, the characterization of the upper semicontinuity of F at � givenin Theorem 5.1.2 does not rely on the coefficients of the nominal problem � .As positive counterpart, it has a straightforward consequence:

Corollary 5.1.1. Let � 2 domF . If F .�/ is bounded, then F is usc at � .

To go ahead, let us observe that the upper semicontinuity of F at � stronglydepends on the coefficients (a and b) as it becomes obvious from Example 3.6 in[104]. This example considers the equivalent representation of the constraint systemof �

� WD fka0t x � kbt ; .t; k/ 2 T � Ng;


whose index set is of the same cardinality than T (when T is infinite), and it showsthat F is trivially usc at � . So, by enlarging the coefficients we forced F to beusc. This observation motivates the introduction in [45] of the reinforced systemassociated with � which is given by

�re WD ˚a0x � b; .a; b/ 2 0C clC.�/�;

and that constitutes the main tool in the analysis of the upper semicontinuity of F .Taking into account the sequential interpretation of the recession directions,

and thanks to the Farkas Lemma, we have 0C clC.�/ � clK.�/, and the set ofsolutions of �re , represented by F re .�/, contains F .�/. We denote by Kre.�/ thecharacteristic cone of �re . It is easy to verify [45, Lemma 3.4] that if d1.�; �/ <1we have �re D �re as 0C clC.�/ D 0C clC.�/, and that Kre.�/ is closed [45,Corollary 5.2].

By using the reinforced system we can establish the following sufficient con-dition [45, Corollary 3.5] for the upper semicontinuity of F when F .�/ is notbounded:

Corollary 5.1.2. Let � 2 domF . If F re.�/ŸF .�/ is bounded, then F is usc at � .

Sketch of the Proof. If � > 0 is such that F re.�/ŸF .�/ � �B2 and d1.�; �/ <1, one has

F.�/ŸF .�/�F re.�/ŸF .�/ D F re.�/ŸF .�/ � �B2,

and F.�/Ÿ�B2�F.�/Ÿ�B2 for all � to a finite distance from � . Now, Theo-rem 5.1.2 applies.

The sufficient condition established in this corollary is not necessary as the sameExample 5.1.1 shows. Remember that for the problem � studied there, F.�/ DŒ�1; 1� � RC and F was usc at � . Moreover, straightforward considerations leadus to

clC.�/ D C.�/ D fy 2 R3 W y2 D 1 and y3 � � jy1jg;

and so,

0C clC.�/ D fz 2 R3 W z2 D 0 and z3 � � jz1jg

D conef.�1; 0;�1/; .1; 0;�1/g:Therefore,

�re D f�x1 � �1; x1 � �1g and F re.�/ D Œ�1; 1� � R;

and it turns out that F re.�/ŸF .�/ is unbounded. Moreover

Kre.�/ D conef.�1; 0;�1/; .1; 0;�1/g: (5.3)


Indeed, according to [45, Theorem 4.5], the last example illustrates the only casewhen the sufficient condition of Corollary 5.1.2 fails to be necessary: 0CF re.�/ DRfug and 0CF.�/ D RCfug for some u ¤ 0n. The following theorem [45, Theorem5.3] covers all the possibilities, and it is also based on the use of the reinforcedsystem.

Theorem 5.1.3 (Upper Semicontinuity of F and the Reinforced System). IfF .�/ is unbounded, two cases are possible:

(i) If F .�/ contains at least one line (i.e., if dimfat ; t 2 T g < n), then F is uscat � if and only if Kre.�/ D clK.�/.

(ii) Otherwise, if w is the sum of a certain basis of Rn contained in fat ; t 2 T g, thenF is usc at � if and only if there exists ˇ 2 R such that

cone .Kre.�/ [ f.w; ˇ/g/ D cone .clK.�/ [ f.w; ˇ/g/ : (5.4)

Let us apply Theorem 5.1.3 to Example 5.1.1. Since F .�/ contains no line, weare in case (ii), and we shall take the basis fa�1; a1g � fat ; t 2 T g. Obviouslyw D a�1 C a1 D .0; 2/ and, according to (5.2) and (5.3), (5.4) holds for ˇ D 0.This allows us to conclude the upper semicontinuity of F at � .

The following mappings are closely related to F W• The boundary mapping B W ˘ � R

n associating with each � 2 ˘ the setB .�/ WD bdF .�/.

• The extreme points set mapping E W ˘ � Rn associating with each � 2 ˘ the

set E .�/ of extreme points of F .�/.Let � 2 domF be such that F .�/ ¤ R

n. The following diagram summarizesthe stability properties of B and the existing relationships with the correspondingproperties of F W

F lsc at � ! B lsc at �&

B closed at �%

F usc at � � B usc at �

The equivalence F lsc at � , B lsc at � means that the latter conditioncould be aggregated to the characterizations of the lower semicontinuity of F inTheorem 5.1.1, under the mild assumption that F .�/ ¤ R

n. Moreover, B lsc at� , B closed at � if dimF .�/ D n and B usc at � , B closed at � if F .�/ isbounded.

The extreme points set mapping E is remarkably unstable unless the followingCQ holds: � is non-degenerate if jT .x/j < n for all x 2 B .�/ŸE .�/ D.bdF .�//Ÿ .extrF .�//. In the next diagram �H WD .c; a; 0/ consists ofreplacing b in � by the null function, so that the constraint system of �H is the


homogeneous system �H WD˚a0t x � 0; t 2 T

�. If jT j � n, E .�/ ¤ ;, and

jF .�/j > 1 (the most difficult case), then one has:

F lsc at � ! E lsc at �#.1/

E closed at �

.4/

�! � non-deg..2/ #" .3/

E usc at �

.5/

�! � & �H non-deg.

The equivalence F lsc at � , E lsc at � gives another condition to beadded to the list in Theorem 5.1.1 under the mild assumptions that jT j � n

(superfluous in LSIO) and F .�/ contains more than one point but not completelines. The implications in the above diagram, with the exception of (5), hold undersome additional assumptions: (1) F .�/ is strictly convex; (2) F .�/ is bounded;(3) fat ; t 2 T g is bounded; and (4) F is lsc at � . The converse statements of (4) and(5) are true if jT j <1.

The known continuity properties of the optimal set mapping S and the optimalvalue function # are gathered in the next two theorems. Through these results thereader can appreciate how strong the influence of the lower semicontinuity of F is.

Theorem 5.1.4 (Stability of the Optimal Set). Let � 2 domS . Then, thefollowing statements hold:

(i) S is closed at � if and only if either F is lsc at � or F .�/ D S .�/.(ii) S is lsc at � if and only if F is lsc at � and S .�/ is a singleton set.

(iii) If S is usc at � , then S is closed at � . The converse is true whenever S .�/ isbounded.

Theorem 5.1.5 (Stability of the Optimal Value). Let � 2 domF . Then, thefollowing statements hold:

(i) If S .�/ is a nonempty compact set, then # is lsc at � . The converse statementholds if # .�/ ¤ �1.

(ii) # is usc at � if and only if F is lsc at � .(iii) If S .�/ is a nonempty compact set and F is lsc at � , then # is Lipschitz

continuous at � , i.e., there exist V 2 N� and L > 0 such that

j# .�1/ � # .�2/j � Ld .�1; �2/ for all �1; �2 2 V:

The latter two results involve the following conditions aside the lower semicon-tinuity of F at � discussed above: (a) S .�/ D F .�/, (b) S .�/ is a nonemptycompact set, and (c) S .�/ is a singleton set. These conditions admit geometricinterpretations in terms of the data although they cannot easily be checked inpractice:


(a) By the non-homogeneous Farkas lemma, S .�/ D F .�/ if and only if.c; # .�// belongs to the lineality of clK .�/.

(b) S .�/ is a nonempty compact set if and only if c 2 intM .�/. A sufficientcondition is the existence of some feasible solution � of the dual problem suchthat span fat ; t 2 � .�/g D R

n.(c) If x 2 F .�/ satisfies c 2 intA.x/, then x is a strongly unique solution of � .

Example 5.1.2. Denote by �1, �2 and �3 the corresponding parameters inExample 1.1.1, with cost vectors c1 D .1; 1/, c2 D .�1;�1/, and c3 D .1; 0/,respectively. Let � be any of these three parameters. Since the SSCQ holds and02 2 int convfat ; t 2 T g, F satisfies all the properties listed in Theorem 5.1.1, inparticular it is lsc at � . Moreover, since the nominal feasible set is compact, F isalso usc at � . Consequently, the boundary mapping B is lsc and closed at � (theupper semicontinuity also holds, but it is not consequence of the general theory).Concerning the extreme point set mapping E , it is lsc at � but F .�/ is not strictlyconvex, so that the upper semicontinuity and closedness of E at � must be justifiedin terms of the data.

Now we apply Theorem 5.1.4. Since F is lsc at � , the optimal set mapping S isclosed at � . This, together with the boundedness of S .�/, guarantees that S is uscat � . The primal problems of �1 and �2 have unique optimal solution, so that S islsc at that parameters while the primal problem of �3 has multiple optimal solutionsand so S fails to be lsc at �3.

Finally, concerning the optimal value mapping # , by Theorem 5.1.5, it is lsc andusc at � as S .�/ is a nonempty compact set and F is lsc at � .

Example 5.1.3 (Uniform Approximation). The best uniform approximation of anuncertain function ' W T ! R, where T is a compact interval in R, by polynomialsof degree less than n can be formulated as an uncertain unconstrained minmaxproblem as follows:

P0 W infx2Rn(

supt2T

ˇˇˇ'.t/ �

nX

iD1t i�1xi

ˇˇˇ

)

:

Introducing a new variable xnC1 WD supt2Tˇ'.t/ �Pn

iD1 t i�1xiˇ, P0 can be

reformulated as an uncertain LSIO problem as follows:

P0 W inf.x;xnC1/2RnC1 xnC1s.t.

ˇ'.t/ �Pn

iD1 t i�1xiˇ � xnC1; t 2 T:

We built up a parametric model for P0 by assuming the existence of a nominalfunction ' W T ! R to be approximated. The corresponding nominal approximationproblem can be formulated as

P' W inf.x;xnC1/2RnC1 xnC1s.t.

PniD1 t i�1xi C xnC1 � ' .t/ ; t 2 T;

�PniD1 t i�1xi C xnC1 � �' .t/ ; t 2 T:


We denote by ˘ the space of arbitrary perturbations of all the data in P' . LetF , S , and # be the corresponding feasible set, optimal set, and optimal valuemappings associated with P' . We are exclusively interested in those perturbationsof the data in P' preserving the coefficient of the variables and such that the RHSfunctions in both blocks of constraints, the result of perturbing ' and�', have zero-sum. In other words, we deal with perturbations consisting of replacing ' W T ! R

in P' with another function ' W T ! R of the same type. So, possible spacesof perturbations are the space ˘1 WD C .T / of real-valued continuous functionson T when ' is continuous, the space ˘2 WD `1 .T / of bounded functions onT when ' is bounded, and the space ˘3 WD R

T when ' is unbounded, with˘1 � ˘2 � ˘3. All these spaces (that can be seen as subspaces of ˘ in anobvious way) are equipped with the supremum metric d1 describing the uniformconvergence on T . If ' 2 ˘1, then P' has a unique optimal solution, i.e., S .'/is a singleton set (see, e.g., [197, Theorem 7.6] or [70]). If ' is unbounded and' 2 ˘3 satisfies d1 .'; '/ < 1, then ' is unbounded too and d1 .'; p/ D C1for any polynomial p, so that P' is inconsistent. This means that F ;;S ;,and # C1 on f' 2 ˘3 W d1 .'; '/ <1g 2 N' . Thus we have just to considerthe continuity properties of the restrictions of F , S , and # to ˘1 and ˘2.

Let ' 2 ˘i , i D 1; 2. The requirement that the admissible perturbations of theRHS function of P' preserve the zero-sum condition obliges to apply carefully theprevious results, which only provide sufficient conditions in the present framework:if one of the relevant mappings, F , S , and # , satisfies a certain stability property at' for arbitrary perturbations, the same property holds for F j˘i , S j˘i , and # j˘i .

Concerning the feasible set mapping, F is closed and lsc at ' because .0n; ı/ is astrong Slater point forP' whenever ı > sup t2T j' .t/j (take " D ı�sup t2T j' .t/j),so that F j˘i is closed and, by Theorem 5.1.1, lsc at ' too. Checking the usc propertyof F j˘i at ' requires ad hoc arguments.

Concerning the optimal set mapping, since F is lsc at ', S is closed at ' (seeTheorem 5.1.4), and it is lsc at ' if S .'/ is a singleton set (as it happens if i D 1).So, the same statements are true for S j˘i .

Finally, concerning the optimal value function, # is usc at ' (see Theorem 5.1.5)and so # j˘i is usc at ' too. Moreover, if S .'/ is a nonempty compact set (e.g.,when ' is continuous), then # j˘i is lsc at ', and it is Lipschitz continuous in someneighborhood of '.

Remark 5.1.1 (Stability of the Feasible Set: Antecedents and Extensions). Thecharacterizations (i)–(vi) and (viii) of the lsc property of F in Theorem 5.1.1 weregiven in [103], and (x) in [101]. All of them, together with an equivalent version of(ix), appear in [102, Theorem 6.1 and Exercise 6.5 for (ix)], and (vii) in [115].

Concerning the upper semicontinuity of F , the sufficiency of the boundedness ofthe feasible set (Corollary 5.1.1) was shown in [104] (also in [102, Corollary 6.2.1])while the characterization given in Theorem 5.1.3 was proved in [45]. The lsc andusc of F for systems of infinitely many convex constraints was studied in [179].

Table 5.1 summarizes the available information on the stability of the feasibleset spread on the references listed at the 1st column (they are chronologicallyordered, with an asterisk marking those works dealing with systems posed in infinite


Table 5.1 Antecedents on the stability of the feasible set

Ref. Year Constr. system lsc usc cl Top c S T

[202]� 1975 sinf. linear cont.[63] 1975 ord. linear X[123] 1975 sinf. X X X[203]� 1976 sinf. C1[222]� 1977 sinf. linear X[29] 1982 sinf. cont. X X X X[82] 1983 sinf. linear cont. X X X[11]� 1983 ord. convex X X X X[31] 1984 sinf. linear cont. X X X X[220] 1985 sinf. linear cont. X[124] 1986 ord. C1 X[131] 1990 sinf. linear cont. X X[152] 1992 sinf. C1 X[183] 1994 ord. linear X[103] 1996 sinf. linear X X X X[101] 1996 sinf. linear X[104] 1997 sinf. linear X[151] 1998 sinf. C1 X[102] 1998 sinf. linear X X X X X X X[177]� 1998 sinf. linear X X X[139] 2000 sinf. linear X[187]� 2000 sinf. linear X X[105] 2001 sinf. linear X[170] 2001 sinf. convex X X X X X[45] 2002 sinf. linear X[154] 2004 sinf. linear X X[46] 2005 sinf. linear X X[4] 2006 sinf. linear X X[3] 2008 sinf. linear X X X[64] 2013 sinf. linear X X

dimensional spaces). For the sake of brevity we do not include information onthe topology defined on the corresponding parameter space. Column 3 informsabout the type of constraint system of P (there “ sinf.,” “ord.,” and “cont.” areabbreviations of “semi-infinite,” “ordinary,” and “continuous,” respectively). Thebinary information (yes or not) of the columns 4–10 is referred to the followingdesirable stability properties of the feasible set:

Column 4: Lower semicontinuity of F at � (lsc).Column 5: Upper semicontinuity of F at � (usc).Column 6: Closedness of F at � (cl).Column 7: Topological stability of F at � (top).Column 8: Continuity of F at � (c).Column 9: Slater-type CQ (S).Column 10: Tuy stability of F at � (T).


Remark 5.1.2 (Hausdorff Semicontinuity of the Feasible Set). Other concepts oflower and upper semicontinuity have been used in [29, 30], and [105] to describethe stability behavior of F W• M is Hausdorff lower semicontinuous (H-lsc in short) at y if for each real

number " > 0 there exists V 2 Ny , such that M .y/ � M .y/ C "B2, forall y 2 V .

• M is Hausdorff upper semicontinuous (H-usc) at y if for each real number " > 0there exists V 2 Ny such that M .y/ �M .y/C "B2, for all y 2 V .

Obviously, Hausdorff lower semicontinuity implies lower semicontinuity andupper semicontinuity implies Hausdorff upper semicontinuity. There is a consensusof experts on the excessive strength of the H-lsc property and the excessive weaknessof the H-usc property in our framework.

Remark 5.1.3 (Stability of F in Problems with Set Constraint and/or Equations).Several papers have been devoted to the qualitative stability analysis of LSIO prob-lems containing linked inequalities to be preserved by any admissible perturbation([3, 4, 36], where each equation can be interpreted as two zero-sum inequalities),problems including a set constraint (e.g., x 2 R

nC when the decision variablessatisfy physical constraints xi � 0, i D 1; : : : ; n/ or both [3, 4]. For instance, inExample 5.1.2 we could consider fixed the sign constraints, in which case the spaceof parameters, say ˘1, is a subset of ˘ . All the properties of F , S W ˘ � R

n and# W ˘ 7! R at � 2 ˘1 are inherited by Fj˘1 , Sj˘1 , and # j˘1 , but the conversestatements are not necessarily true. So, in this particular case, where S fails to be lscat �3, we should determine whether Sj˘1 is lsc or not at �3 (in fact it is not becausearbitrarily small perturbations of c3 convert the optimal set into a singleton set).

Remark 5.1.4 (Application to Voronoi Cells). Linear semi-infinite systems arise indifferent branches of mathematics as convex analysis (e.g., the convex and theconcave subdifferentials are solutions sets of such type of systems), robust linearcomplementarity problems [229, Definition 1.10], or computational geometry. Themetric projection on T � R

n, jT j � 2, is the set-valued mapping PT W Rn � Rn

associating with each x 2 Rn the set of nearest points in T for the Euclidean

distance. So, given s 2 T; P�1T .s/ represents the set of all points of R

n closerto s than to any other element of T . In this framework, the elements of T are calledVoronoi sites while P�1

T .s/ is the Voronoi cell of s. Until the 1930s, only finitesets of sites were considered, e.g., by Descartes in 1644, Dirichlet in 1850, andVoronoi in 1908, for n D 2, n D 3, and n > 3, respectively. Delaunay published in1934 a paper on crystallography where he considered a discrete infinite set T andn 2 N. Voronoi cells for finite sets are widely applied in computational geometry,operations research, data compression, economics, marketing, etc. As observed in[226], eliminating kxk22 in the 2nd inequality of

d2 .x; s/ � d2 .x; t/, kx � sk22 � kx � tk22 for all t 2 T;


one gets

P�1T .s/ D fx 2 R

n W d2 .x; s/ � d2 .x; t/ ; t 2 T gDnx 2 R

n W .t � s/0 x � ktk22�ksk222

; t 2 To:

This reformulation has been systematically exploited in [108] and [109] to getgeometric information on P�1

T .s/ from the data (T and s 2 T ) and to determinethose sets T such that P�1

T .s/ is a given closed convex set, respectively. The effectof different types of perturbations of the nominal data, a couple

�T ; s

�such that

s 2 T � Rn, on the Voronoi cells has been analyzed in [116].

Remark 5.1.5 (Primal-Dual Stability). In the same way that int domF can beinterpreted as the set of primal stable consistent parameters (in the sense thatsufficiently small perturbations provide primal consistent problems), the topologicalinterior of the main subsets of ˘ can be seen as the sets of stable parameters inthe corresponding sense. Some of these interiors have been characterized in thecontinuous case [111, 113] and the general case [191], e.g., those corresponding tothe partitions (inconsistent-bounded-unbounded or inconsistent-solvable-boundedunsolvable-unbounded) corresponding to the primal problem, the dual problem, orboth problems.

Remark 5.1.6 (Stability of the Boundary and the Extreme Points Set). The resultson the stability of the boundary summarized in the diagram after Theorem 5.1.3can be found in [99, 106]. In the latter paper, the relationships between thestability properties of set-valued mappings with closed convex images in R

n andtheir corresponding boundary mappings have been analyzed in full generality. Theequivalence “B closed at � , B usc at �” was used in [94] in order to obtain asufficient condition for the stable containment between solution sets of linear semi-infinite systems.

The seminal results on the Hausdorff semicontinuity of the set of extreme pointsof the feasible set of finite systems in [66] were first extended to the feasible setof infinite systems in [100] and then to arbitrary set-valued mappings with conveximages in [114].

Remark 5.1.7 (Stability of S and # W Antecedents and Extensions). Table 5.2reviews briefly a non-exhaustive list of relevant works, chronologically ordered, onstability of optimization problems of the form

inf f .x; �/ s.t. x 2 F .�/ ;

where typically,

F.�/ D fx 2 X W ft .x/ � 0;8t 2 T I x 2 C g;


Table 5.2 Antecedents on the stability of the optimal set

Ref. Year X T f f .t; �/ f .�; x/ K # S[136] 1973 top abstr lsc or usc – – – X X[29] 1982 R

n top lin fract arb RT� X X

[11] 1983 met abstr lsc or usc – – – X X[30]� 1983 R

n compH cont aff cont RT� X

[82] 1983 Rn compH lin aff cont R

T� X

[31] 1984 Rn compH lin aff cont R

T� X X

[62] 1984 norm compH cont aff cont RT� X

[157] 1985 Rn fin fin conv fin conv/aff – R

T� X X

[10]� 1997 met abstr lsc – – – X[102] 1998 R

n arb lin aff arb RT� X X

[156] 1998 Rn compH diff diff cont R

T� – X

[25] 2000 Ban arb cont cont – cl conv X X[48] 2001 R


[86] 2003 Rn arb fin conv fin conv arb R

T� X X

[145] 2005 Ban fin fin conv fin conv/aff – cl conv X X[47] 2005 R


[180]� 2006 met abstr fin usc – – – X[53] 2006 R

n arb lin aff arb RT� X

[43]� 2007 Rn met compH fin conv fin conv cont R

T� X

[72]� 2007 lcH arb lsc conv lsc conv arb RT� X

[143]� 2011 Rn compH fin conv fin conv cont R

T� X

[73] 2012 Ban arb lsc lsc arb RT� X X

X denoting the decision space, C denotes a fixed set constraint C (generallyC D X), and � 2 ˘ (the corresponding parameter space). Those works dealingwith particular types of perturbations, usually right-hand side (RHS) perturbations,are marked with an asterisk. For comparison purposes, we represent here thefunctional constraints as f .t; �/ 2 K, where f .t; x/ WD ft .x/ and K is a givensubset of certain partially ordered space Y (e.g., Y D R

T and K D RTC for our

LSIO problem P ). We codify the information in the columns 3–8 of Table 5.2 asfollows:

Column 3: Banach (Ban), normed (nor), metric (met), locally convex Hausdorfftopological vector space (lcH), and topological space (top).Column 4: finite (fin), arbitrary (arb), and compact Hausdorff topological space(compH). In case of abstract minimization problems (abstr), there is no explicitinformation on T , f .t; �/, f .�; x/, and K.Columns 5–7: affine (aff), linear (lin), fractional (fract), convex (conv), finite val-ued (fin), continuous (cont), lower semicontinuous (lsc), upper semicontinuous(usc), arbitrary (arb), and continuously differentiable (diff). In case of abstractminimization problems, no direct information on the constraints is available andthe usual allowed perturbations are sequential.

5.2 Stability Restricted to the Domain of the Feasible Set 73

Column 8: closed (cl) and convex (conv). For the sake of brevity we do notinclude in this table information on the parameter space and the only stabilityconcepts considered here are exclusively lower and upper semicontinuity andclosedness. This precludes, among other stability concepts related to S , theLipschitzian and Hölder stabilities [25], the structural stability [151, 153] or thestability of stationary solutions [133].

In [156], T changes with the parameter, but it is always compact and uniformlybounded. Reference [145] deals with the well-posedness of convex programs underlinear perturbations of the objective functions and RHS perturbations.

5.2 Stability Restricted to the Domain of the Feasible Set

A new framework has been suggested by the stability theory of zero sum games,where the perturbations are forced to preserve the feasibility of the involved systems[181]. Thus, the challenge consists of characterizing the lsc property of F restrictedto domF , say FR, for different types of systems, under perturbations of all thedata, of the RHS function, and of the LHS function. Let us mention two resultsconcerning the latter type of perturbations (i.e., when only a is perturbed, and b andc remain fixed), for arbitrary and continuous systems, which constitute a novelty inliterature.

Theorem 5.2.1 (Lower Semicontinuity of FR). Let � D .c; a; b/ be an ordinaryLSIO, and consider only arbitrary perturbations of a. Then the following statementsare true:

(i) If either � satisfies SSCQ or F .�/ is a singleton set, then FR is lsc at � .(ii) If FR is lsc at � 2 int domF and F.�/ ¤ f0ng, then � satisfies the SSCQ.

(iii) If FR is lsc at � and F.�/ is neither a singleton set nor a subset of a ray, then� satisfies the SSCQ.

Theorem 5.2.2 (Dimensional Stability of FR). Let � D .c; a; b/ 2 domF be acontinuous LSIO and consider only continuous perturbations of a. Suppose that �has not trivial inequalities as constraints and that 0n … F.�/: Then, the followingstatements are equivalent to each other:

(i) There exists V 2 N� such that dimFR .�/ D dimFR .�/ for all � 2 V .(ii) SSCQ holds.

(iii) dimF.�/ D n.

Moreover, any of these properties implies that FR is lsc at � .

The remarkable difference between Theorem 5.1.1 and Theorems 5.2.1 and 5.2.2is due to the systems in bd domF (as FR and F coincide on int domF). In theparticular case when T is finite, conditions (i)–(iii) in Theorem 5.2.2 hold if andonly if one of the following alternatives holds:


(a) dim F.�/ D nI(b) dimF.�/ D 0I(c) F.�/ is a non-singleton set contained in some open ray (a half-line emanating

from 0n without its apex).

Remark 5.2.1 (Antecedents and Sources). Theorems 5.2.1 and 5.2.2 are [64, Propo-sition 9] and [64, Proposition 10], respectively.

5.3 Well and Ill-Posedness

The LSIO problem P is said to be well-posed (ill-posed) w.r.t. a certain propertywhen this property is satisfied by any perturbed problem provided that the pertur-bation is sufficiently small (arbitrarily small perturbations of P provide problemssatisfying or not that property, respectively). In topological terms, P is said to bewell-posed (ill-posed) when its associated parameter � D .c; a; b/ is an interiorpoint (a boundary point, respectively) of the set of parameters corresponding toproblems satisfying such a property. For instance, P is well-posed (ill-posed) withrespect to primal feasibility when � 2 int domF (� 2 bd domF , respectively).

The following subspaces of ˘ are relevant in the primal/dual analysis ofwell-posedness:

˘c D domF ; ˘Dc D domFD;

˘i D ˘Ÿ˘c; ˘Di D ˘Ÿ˘D

c ;

˘s D domS; ˘Ds D domSD;

˘b D f� 2 ˘c W # .�/ > �1g;˘Db D f� 2 ˘D

c W #D .�/ > �1g;˘u D ˘cŸ˘b; ˘D

u D ˘Dc Ÿ˘D

b :

Obviously the subspaces ˘i;˘b , and ˘u (˘Di ;˘

Db , and ˘D

u ) constitute a partitionof ˘ , called primal partition (dual partition, respectively).

The following theorem accounts for some results about well-posedness(ill-posedness), confined to the primal setting of general LSIO (not necessarilycontinuous). Aside the sets C.�/; M.�/, N.�/, and K.�/, the statements belowinvolve the following sets associated with � D .c; a; b/ W

A.�/ WD convfat ; t 2 T g; H.�/ WD C.�/C RC.0n;�1/;

ZC.�/ WD convfat ; t 2 T I cg; and Z�.�/ WD convfat ; t 2 T I �cg:

Obviously, M.�/ D RCA.�/.

5.3 Well and Ill-Posedness 75

Theorem 5.3.1 (Primal Well-Posedness). Given � D .c; a; b/ 2 ˘ , thefollowing statements hold:

(i) If d.�; bd˘c/ < C1, then � 2 int˘i , � 2 int˘c , or � 2 bd˘c if and only if0nC1 2 intH.�/, 0nC1 2 int.RnC1ŸH.�//, or 0nC1 2 bdH.�/, respectively.

(ii) If � 2 int˘c , then � 2 int.˘cŸ˘s/, � 2 int˘s , or � 2 bd˘s if and only if0n 2 int.RnŸZ�.�//, 0n 2 intZ�.�/, or 0n 2 bdZ�.�/, respectively.

(iii) cl˘s D cl˘b and int˘s D int˘b (hence bd˘s D bd˘b/.

In (i) we are excluding those problems � such that d.�; bd˘c/ D C1,problems whose existence and properties are studied in the following chapter. Inparticular we will see that these problems are inconsistent.

Example 5.3.1. Consider the problem, in R,

� W infx2R.�x/ s.t. 0x � 1; x � �k, k 2 N.

This problem is obviously inconsistent, but � 2 bd˘c as the problems

�" W infx2R.�x/ s.t. "x � 1; x � �k, k 2 N,

with " > 0 are consistent, and �" ! � as " # 0. Therefore, statement (i) aboveapplies (certainly, the reader may verify that 02 2 bdH.�/).

Example 5.3.2. Let � be any of the three parameters �1, �2 and �3 considered inExample 1.1.1. We already know that � 2 int˘c and it is easy to check that 0nC1 2int.RnC1ŸH.�//; in fact, d2.0nC1;H.�// D .5 C 2p2/�2 (see Example 6.2.1).Since 02 2 intA.�/ � intZ�.�/ we also conclude that � 2 int˘s .

It makes sense to call totally ill-posed problems to those problems in .bd˘c/ \.bd˘s/, since they are simultaneously ill-posed with respect to both feasibilityand solvability. The following characterization of these problems does not involveexclusively the data (so, it is hard to be checked). Let us observe that � 2 bd˘c

entails either � 2 bd˘s or � 2 int.˘Ÿ˘s/.

Theorem 5.3.2 (Total Ill-Posedness). If � 2 bd˘c , then

� 2 bd˘s , either 0n 2 bdZC .�/ or � 2 cl.˘c \ bd˘c/: (5.5)

Example 5.3.3. In Example 5.3.1, it is easy to see that in a sufficiently smallneighborhood of � , any problem � is either inconsistent or unbounded, i.e., � 2int.˘Ÿ˘s/: In fact 0 2 intZC .�/ D Œ�1; 1� and � … cl.˘c \ bd˘c/, since anyconsistent problem close enough to � satisfies 02 2 int

�R2ŸH.�/

�, and hence

belongs to int˘c .

In the continuous setting we have the following primal-dual results:


Theorem 5.3.3 (Primal-Dual Well-Posedness). In the parametric space of con-tinuous problems ˘ , the following results hold for � D .c; a; b/ 2 ˘ :

(i) � 2 int˘c , � satisfies the SCQ.(ii) � 2 int˘D

c , c 2 intM.�/.(iii) � 2 int˘b , � 2 int

�˘b \˘D

b

� , � 2 int˘Db , � satisfies the SCQ

and c 2 intM.�/.(iv) � 2 int˘i , � 2 int

�˘i \˘D

u

�, � 2 int˘Du , .0n; 1/ 2 intK.�/.

(v) � 2 int˘Di , � 2 int

�˘u \˘D

i

� , � 2 int˘u , 9y 2 Rn such that

c0y < 0 and a0t y > 0 for all t 2 T .

(vi) int�˘i \˘D

i

� D int�˘b \˘D

i

� D int�˘i \˘D

b

� D ;.The condition in (iii), i.e., � satisfies SCQ and c 2 intM.�/, also characterizes

well-posedness w.r.t. simultaneous boundedness (i.e., � 2 int�˘b \˘D

b

�) for a

general LSIO, not necessarily continuous. On the contrary, and according to (vi),no continuous LSIO problem is well-posed w.r.t. simultaneous infeasibility, while ageneral LSIO problem � is well-posed w.r.t. the same property if and only if

0n … clA.�/; c … clM.�/; and .0n; 1/ 2 0C clC.�/:

Remark 5.3.1 (Sources and Related Results). Statement (i) in Theorem 5.3.1 comesfrom Theorems 4 and 5 in [49], (ii) and (iii) are Theorem 2 and Theorem 1 in[51], respectively. Theorem 5.3.2 can be found in [51, Theorem 3], and sufficientconditions for total ill-posedness are established in [54]. In [127], condition (5.5) ischaracterized in terms of the data, by using the formula for the subdifferential of thesupremum function given in [128]. Example 5.3.1 is Example 1 in [51].

In [221] Lipschitz constants for both primal and dual optimal value functionsare derived under weaker assumptions of stability, which do not preclude, in all thecases, the existence of duality gap. The allowed perturbations are restricted to thecoefficients of the objective function of the corresponding dual problems.

The well-posedness results on continuous LSIO problems given in Theorem 5.3.3come from [111, 113]. These results have been extended to general LSIO in [191].Moreover, the well-posedness w.r.t. the existence of a strongly unique optimalsolution of the primal problem has been characterized in [115]. The mentionedcharacterizations of well-posedness, always in terms of the data, may not coincidefor both continuous and general LSIO.

Refined results for the case in which ˘b and ˘Db are split into sets composed

by parameters which have compact optimal sets and those for which this desirableproperty fails are given in [112]. Moreover, in [112] it is also shown that mostparameters having either primal or dual bounded associated problems have primaland dual compact optimal sets. This generic property fails for general problems (notcontinuous), despite almost all the characterizations of the topological interior ofabove subspaces of ˘ are still valid for the general LSIO problem [50, 52, 105].

Formulas in terms of the data to compute (or at least to estimate) the distancefrom a given well-posed problem to ill-posedness w.r.t. primal feasibility, infeasi-bility, solvability and unsolvability (i.e., the minimum size of those perturbations

5.3 Well and Ill-Posedness 77

which provide problems not satisfying the corresponding property) have beenprovided by Cánovas, Hantoute, López, Parra, and Toledo in a series of paperspublished from 2005 until 2008 [49–52, 127].

Remark 5.3.2 (Some Open Problems in Qualitative Stability of LSIO Problems).

1. Theorem 5.1.1 and other results above provide a long list of properties which areequivalent to the lower semicontinuity of F under perturbations of data whichinclude b. It remains to determine, from these properties, which still characterizethe lower semicontinuity of F under perturbations which keep b fixed.

2. Theorem 5.1.3 characterizes the upper semicontinuity of F under perturbationsof the triple, but there are not counterparts of this result for other types ofperturbations of F , e.g., those which keep b fixed.

Chapter 6Quantitative Stability Analysis

6.1 Quantitative Stability of Set-Valued Mappings

The following stability properties of a set-valued mapping are of quantitative nature.Again M W Y � X is a set-valued mapping between two spaces Y andX equippedwith (possible extended) distances denoted by d .

• M is pseudo-Lipschitz at .y; x/ 2 gphM if there exist V 2 Ny , U 2 Nx , and ascalar � 0 such that

d .x;M .y// � d �y; y0� 8y; y0 2 V; 8x 2M.y0/ \ U: (6.1)

Observe that this is a stable property, in the sense that if M is pseudo-Lipschitz at.y; x/, it is also pseudo-Lipschitz at points of the graph around .y; x/. The pseudo-Lipschitz of M at .y; x/ is also known as the Aubin continuity or Lipschitz-likeproperty of the mapping (see [158, Sect. 1.4] and references therein for details).Moreover, the pseudo-Lipschitz of M at .y; x/ turns out to be equivalent to themetric regularity of the inverse M�1 at .x; y/ I i.e., to the existence of V 2 Ny ,U 2 Nx , and a scalar � 0 such that

d .x;M .y// � d �y;M�1 .x/� 8y 2 V; 8x 2 U: (6.2)

Indeed, (6.2) trivially implies (6.1), while, assuming (6.1), it can be proved that forsmaller neighborhoods we may remove y0 2 V , requiring only y0 2 M�1 .x/.Formally, starting from (6.1), one derives (6.2) with the same constant andpossibly smaller neighborhoods U and V (see again [158, Sect. 1.4]).

The situation is illustrated in Fig. 6.1, where one observes that, according to (6.2),the smaller this bound is, the more stability of the image sets M .y/ is held aroundy. The infimum of such for all the triplets . ; U; V / verifying (6.2), which doescoincide with the infimum of for all . ; U; V / in (6.1), is called exact Lipschitzian


79

80 6 Quantitative Stability Analysis

Fig. 6.1 Image sets and pseudo-Lipschitz property

bound (or Lipschitz modulus) of M at .y; x/ and it is denoted by lip M .y; x/ I so,from (6.1) we easily obtain

lip M .y; x/ D lim supy;y0! Ny; y¤y0

x! Nx; x2M.y/

d .x;M .y0//d .y; y0/

:

(Observe that y and y0 in (6.1) are interchangeable.)If lip M .y; x/ D C1, M fails to be pseudo-Lipschitz at .y; x/.The pseudo-Lipschitz of M at .y; x/ implies the inner semicontinuity of M at

.y; x/ [172, Lemma 2.3]. In fact, the inequality (6.1) for y0 D y gives

d .x;M .y// � d .y; y/ , for every y 2 V: (6.3)

Then, for V in (6.3) and assuming the nontrivial case > 0, if W is an open setcontaining x, we shall take � > 0 small enough to ensure

.x; y/ 2 W � V; whenever d .x; x/ < � and d .y; y/ < �= :

So, appealing to (6.3) one has

W \M .y/ ¤ ;; whenever d .y; y/ < �= ;

and this accounts for the inner semicontinuity of M at .y; x/:If both spaces Y and X are normed and M has closed values, i.e., the images

M.y/ are closed sets in X , (6.1) is equivalent (with the same constant and thesame neighborhoods U and V ) to

M.y0/ \ U �M .y/C ��y � y0��BX; (6.4)

6.1 Quantitative Stability of Set-Valued Mappings 81

Fig. 6.2 Lipschitz continuityof f .y/ D p

pjyj C .1=16/

where k:k is the norm defined on Y and BX is the closed unit ball in X . Without theclosedness assumption, (6.4) still implies (6.1), whereas (6.1) implies (6.4) for any 0 > :

If M f is single valued, (6.1) gives rise to

d�f .y0/; f .y/

� � d �y; y0� 8y; y0 2 V ,

in other words, f is Lipschitz continuous around y (or, equivalently, strictlycontinuous at y).

Example 6.1.1. In Fig. 6.2 we represent the graph of the function f W R!R

defined by

f .y/ Dpjyj C .1=16/

which is obviously Lipschitz continuous at any point (why?).

If f W Rn! Rm is of class C1 on an open set W � R

n, then f is Lipschitz-continuous around every y 2 W , and

lip f .y/ D krf .y/k2 ;

where rf .y/ is the Jacobian matrix of f at y (see, for instance, [207,Theorem 8.7]). Difficulties arise whenever f is not differentiable (or not of classC1/. In Fig. 6.2, the reader may verify by a simple geometrical observation thatlip f .0/ D 2.

In Fig. 6.3 we show a mapping M W R � R which fails to be pseudo-Lipschitzat a particular point .y; x/ 2 gphM. This is why, if we approach y from the rightby points y; y0, we see that the ratio d .x;M .y0// =d .y; y0/ tends to infinity as theright slope of the upper-boundary of graphM isC1.

• M is calm at .y; x/ 2 gphM if there exist V 2 Ny , U 2 Nx , and a scalar � 0such that


Fig. 6.3 Pseudo-Lipschitz property fails

d .x;M .y// � d .y; y/ 8y 2 V; 8x 2M.y/ \ U: (6.5)

Note that M .y/ could be empty for some y 2 V .Equivalently, the calmness property of M can be established in terms of metric

subregularity of M�1 [75], which reads as the existence of a (possibly smaller)neighborhood U of x such that

d .x;M .y// � d �y;M�1 .x/�

, 8x 2 U: (6.6)

The constant is also known as local error bound of d�y;M�1 .�/� at x,

provided that gphM is locally closed around .y; x/.The infimum of such for all the couples . ; U / verifying (6.6) is called

exact calmness bound (or calmness modulus) of M at .y; x/ and is denoted byclmM .y; x/. From (6.5) we get

clmM .y; x/ D lim supy! Ny; x! Nxx2M.y/

d .x;M .y//

d .y; y/: (6.7)

Observe that (6.6) comes from (6.2) by fixing y at y, and so, clmM .y; x/ �lipM .y; x/. Consequently, pseudo-Lipschitz at .y; x/ implies calmness at the samepoint, but calmness at .y; x/ does not imply calmness around the point.

• M is said to be isolatedly calm (or locally upper Lipschitz) if, additionallyto (6.6), M .y/ D fxg.If M f is single-valued, (6.6) becomes

d .f .y/ ; f .y// � d.y; y/, 8y 2 V ,

6.1 Quantitative Stability of Set-Valued Mappings 83

Fig. 6.4 Calm, but notLipschitz

where V is a certain neighborhood of y. Moreover,

clm f .y/ D lim supy! Ny

d .f .y/ ; f .y//

d .y; y/: (6.8)

Next we give some examples to compare calmness and Lipschitz continuity.

Example 6.1.2. In Fig. 6.4 we represent the graph of the function

f .y/ D(y sin 1

y; if y ¤ 0;

0; if y D 0:

We have, from (6.8)

clm f .0/ D lim supy!0

d .f .y/ ; f .0//

d .y; 0/D lim sup

y!0

ˇˇˇsin

1

y

ˇˇˇ D 1:

(Take the sequence yk D 1=.2k� C .�=2//, k D 1; 2; : : :)On the other hand, if we take a new sequence y0

k D 1=.2k� C .3�=2//,k D 1; 2; : : :, also converging to 0, we can write

lipf .0/ D lim supy;y0!0

d .f .y/ ; f .y0//d .y; y0/

� limk!1

d�f .yk/ ; f

�y0k

��ˇyk � y0

k

ˇ

D limk!1.4k C 2/ D C1;

i.e., lipf .0/ D C1, entailing that f is calm at 0 but it is not Lipschitz continuousat this point.


Fig. 6.5 Calm and Lipschitzat y D 0

Example 6.1.3. The function in Fig. 6.5

f .y/ D(y2 sin 1

y; if y ¤ 0;

0; if y D 0;

is not of class C1 at 0 because

f 0.y/ D(2y sin 1

y� cos 1

y; if y ¤ 0;

0; if y D 0;

and limy!0 f0.y/ does not exist.

The reader will verify easily that clm f .0/ D 0. In order to calculate lip f .0/we make the following considerations. For y ¤ y0, the mean-value theorem (it doesnot require that f 2 C1) yields the existence of z 2�y; y0Œ such that

jf .y0/ � f .y/jjy0 � yj D ˇf 0.z/

ˇ;

and, for every " > 0, with y and y0 close enough to 0 we can ensure that

jf .y0/ � f .y/jjy0 � yj D ˇ

f 0.z/ˇ

� j2z sin.1=z/ � cos.1=z/j � j2z sin.1=z/j C jcos.1=z/j � "C 1:

(Remember that f 0.0/ D 0:/ Therefore,

lipf .0/ D lim supy;y0!0

jf .y0/ � f .y/jjy0 � yj � 1:

6.2 Quantitative Stability of the Feasible Set Mapping 85

Moreover, if

yk D 1

2k�; k D 1; 2; : : : ;

we have f 0.yk/ D �1, and by the definition of derivative, there will exist, for eachk, a real number y0

k such that

ˇˇˇ

ˇf .y0

k/ � f .yk/ˇ

ˇy0k � yk

ˇ � ˇf 0.yk/ˇˇˇˇDˇˇˇ

ˇf .y0

k/ � f .yk/ˇ

ˇy0k � yk

ˇ � 1ˇˇˇ� 1

k:

Taking limits above, for k !1,

limk!1

ˇf .y0

k/ � f .yk/ˇ

ˇy0k � yk

ˇ D 1;

and so lipf .0/ D 1.

In this chapter some results on the pseudo-Lipschitz property and the calmnessof F and S are established, and estimates of the corresponding modulus are alsoprovided.

• A global error bound of d�y;M�1 .�/�, provided that gphM is closed, is any

nonnegative scalar < C1 such that

d .x;M .y// � d �y;M�1 .x/�

, for all x 2 X: (6.9)

If such a scalar exists, then the following finite supremum (which coincideswith the infimum of the 0s satisfying (6.9)) constitutes a kind of condition ratefor � W

0 � �.M; y/ WD supx2X

d .x;M .y//

d .y;M�1 .x//;

under the convention 0=0 D 0.

6.2 Quantitative Stability of the Feasible Set Mapping

The first quantitative stability criterion considered in this section is the distance toill-posedness. This concept was introduced, in the context of conic linear systems,by Renegar in [200]. Besides constituting itself a quantitative measure of thestability (and well-posedness) of the system, this distance becomes a key tool in theanalysis of the complexity of certain algorithms, as interior point methods in [201],


the ellipsoid algorithm in [84], or a generalization of von Neumann’s algorithmstudied in [78]. In different subsections we give expressions for d .�; bd domF/,d.�; bd domS/, etc.

6.2.1 Distance to Ill-Posedness with Respect to Consistency

Concerning the estimation of distances to ill-posedness, in a series of papers([49–52], etc.) various formulas were provided to translate the calculus of pseu-dodistances from � D .c; a; b/ to the sets of ill-posed problems into the calculusof distances from the origin to a suitable set in the Euclidean space (Rn or RnC1).We consider in this subsection arbitrary norms in R

n and RnC1, and their associated

extended distances d in the parameter space ˘ given by (2.4). In [49, Sect. 5] it isproved that

d .�; bd domF/ Dˇˇˇ

supx2Rn

inft2T

a0t x � bt

k.x;�1/k�

ˇˇˇ:

In the same paper an alternative expression for d .�; bd domF/ is given, with aclear geometrical meaning. A first difficulty we face here is that, in our framework,it could happen that d .�; bd domF/ D1. For instance, if the constraint system of� is � WD ˚ 1

rx � r; r 2 N

�, one can easily check that

d .�; bd domF/ Dˇˇsupx2R

infr2N

.1=r/x � rk.x;�1/k�

ˇˇ D j�1j D 1:

This situation is only possible when the set C.�/ is unbounded. Observe also thatsuch a system � must be inconsistent. Otherwise, i.e., if � 2 domF , we can obtainan inconsistent system from it just by replacing an arbitrarily chosen constraint of� , for instance a0

t0x � bt0 for a certain t0 2 T , by 00

nx � 1, and so

d .�; bd domF/ D d .�; bd.˘Ÿ domF//D d .�;˘Ÿ domF/ �

��.at0 ; bt0 / � .0n; 1/�� ;

the second equality coming from [49, Corollary 1].Aside the marginal function g, we also make use of the function f W RnC1 !

R [ fC1g, given by

f .x; �/ WD supfbt�C a0t x W t 2 T g: (6.10)

Obviously, f .�x; 1/ D g.x/. The function f is the support function of clC.�/and it is a lower semicontinuous sublinear function, whose effective domain satisfies[134, Proposition V.2.2.4]


cl.dom f / D .0C clC.�//ı:

The subdifferential of f at .x; �/ 2 dom f is

@f .x; �/ D f.u; �/ 2 clC.�/ W f .x; �/ D hu; xi C ��g:

In particular

@f .0nC1/ D clC.�/:

Given �r D .cr ; ar ; br / and defining

fr.x; �/ WD supfbrt �C .art /0x W t 2 T g;

one has for every .x; �/ 2 RnC1 and r D 1; 2; : : :

ˇˇf .x; �/ � fr.x; �/

ˇˇ � d.�r ; �/k.x; �/k�, (6.11)

and d.�r ; �/ < C1, r D 1; 2; : : :, will imply dom f D dom fr . Moreover,if limr!1 d.�r ; �/ D 0, the sequence fr ; r D 1; 2; : : :, will converge to f

pointwisely.The following theorem provides a couple of characterizations of these abnormal

systems.

Theorem 6.2.1 (Systems Such That d1 .�;bd domF/ D C1). The followingstatements are equivalent:

(i) d1 .�; bd domF/ D C1I(ii) the marginal function of � , g.x/ WD supt2T

nbt � a0

t xo, is improper, i.e.,

g D C1I(iii) .0n; 1/ 2 0C clC.�/.

Let us sketch the proof of the equivalence of (i) and (ii). Suppose first thatd1 .�; bd domF/ D C1 and, reasoning by contradiction, assume that g.x0/ <C1. Then, if we consider the perturbed system � D .c; a; b C b/, where b is theconstant function bt D �g.x0/, it is obvious that � 2 domF (as x0 2 F.�/), andwe get the contradiction d1 .�; bd domF/ � d1 .�; �/ D jg.x0/j < C1.

On the other hand, if g D C1 and we consider � D .c; a; b/ such thatd1 .�; �/ is finite, and we define

f .x; �/ WD supfbt�C a0t x W t 2 T g;


then (6.11) yields for any x 2 Rn

g.x/ D f .�x; 1/ � f .�x; 1/�d1 .�; �/ k.x; �/k1 D g.x/�d1 .�; �/ k.x; �/k1and g D C1, entailing � … domF .

To show the equivalence of (ii) and (iii) we use the function f defined in (6.10).Assume that (iii) holds, that is, for every fixed .u; �/ 2 clC.�/ and all � � 0, wehave

.u; �/C �.0n; 1/ 2 clC.�/ D @f .0nC1/:

Thus, for each x 2 Rn,

f .�x; 1/ � ..u; �/C �.0n; 1//0 .�x; 1/ D �C � � hu; xi;

for all � � 0, i.e., g.x/ D f .�x; 1/ D C1.Conversely, assume that (ii) holds but .0n; 1/ … 0C clC.�/. By the separation

theorem, there will exist .v; ˛/ 2 RnC1Ÿf0nC1g and a scalar ˇ such that

.v; ˛/0.z; �/ � ˇ < ˛ for all .z; �/ 2 0C clC.�/:

Since 0C clC.�/ is a closed convex cone, we see from the previous inequalities thatˇ D 0, ˛ > 0, and

.v; ˛/ 2 Œ0C clC.�/�ı D cl.dom f /:

Consequently, applying Theorem 6.1 in [205], and taking into account that f ispositively homogeneous, there would exist x satisfying f .�x; 1/ D g.x/ < C1,and this contradicts (ii).

In Sect. 3.1 we introduced the hypographical set H .�/ WD C .�/ CRC f.0n;�1/g. Observe that

H .�/ D RnC1 ) d1 .�; bd domF/ D C1: (6.12)

Actually this is a straightforward consequence of Theorem 6.2.1 and the fact that

g.x/ D sup.a;b/2H.�/

˚b � a0x

�:

Theorem 6.2.2 (Distance to Ill-Posedness w.r.t. Consistency). If d.�; bd domF/< C1, we have

d .�; bd domF/ D d .0nC1; bdH .�// : (6.13)


Equation (6.13) is valid for an arbitrary norm in RnC1, k�k, and the associated

distance d.�; �/ in the parameter space (see (2.4)).

Example 6.2.1. We are revisiting Example 1.1.1. Any problem � considered there(i.e., for any c 2 R

n) belongs to int domF (SCQ holds) and the reader may verifythat

d2 .�; bd domF/ D .5C 2p2/�1=2;

and that the perturbed problem � D .c; a; b/ where

.at ; bt / WD .at ; bt / � u; t 2 T;

with

u D .5C 2p2/�1.1; 1;�1 �p2/;

is such that � 2 bd domF and d2 .�; �/ D d2 .�; bd domF/. Observe that, in fact,

1Cp25C 2p2.a�=4; b�=4/C

4Cp22.5C 2p2/..a2; b2/C .a3; b3// D 03;

and 03 2 bdH .�/.

6.2.2 Pseudo-Lipschitz Property of F

If we translate the corresponding definition into our context, it turns out by (6.2) thatF W ˘ � X is pseudo-Lipschitz at .�; x/ 2 gphF if there exist V 2 N� , U 2 Nx ,and a scalar � 0 such that

d2 .x;F .�// � d1��;F�1 .x/

� 8� 2 V , 8x 2 U , (6.14)

and the associated Lipschitz modulus is

lipF .�; x/ D lim sup.x;�/!.x;�/

d2 .x;F .�//d1 .�;F�1 .x//

;

under the convention 0=0 D 0.This property has important consequences in the overall stability of the constraint

system of � , as well as in its sensitivity analysis, and affects even the numericalcomplexity of the algorithms conceived for finding a solution of the system. Manyauthors explored the relationship of this property with standard constraint quali-fications as Mangasarian–Fromovitz CQ, SCQ, Robinson CQ, etc. For instance,


in [156] the relationships among metric regularity, metric regularity with respectto right-hand side perturbations, and the extended Mangasarian–Fromovitz CQ areestablished in a non-convex differentiable framework.

Equation (6.14) means that the distance d2 .x;F .�// is bounded from aboveby d1

��;F�1 .x/

�around .�; x/, and this fact is especially useful because

d1��;F�1 .x/

�is a kind of residual, easily computable in general.

To illustrate this difference, let us consider the simplest model of a continuousnominal problem � D .c; a; b/ with continuous perturbations of the right-hand sidefunction b, i.e., we are considering only parameters of the form � D .c; a; b C b/,with associated constraint system

fa0t x � bt C bt ; t 2 T g;

where perturbations b belong to C .T /. In this case the residual accounts for thesupremum of constraint violations of � by x, i.e.,

d1.�;F�1 .x// D maxt2T

hbt C bt � a0

t xi

C ;

whereas to calculate d2.x;F .�// D infx02F.�/ kx � x0k2 is much more compli-cated. In fact, if � has a Slater point, we get the extended Ascoli formula [58]

d2.x;F .�// D max.a;˛/2C.�/

Œ˛ � a0x�Ckak2

;

with Œ��C denoting the positive part, and C.�/ WD convf.at ; bt C bt / W t 2 T g.Let us sketch the proof of this result as it comes by a straightforward application

of our standard tools. In fact we prove that, for every x 2 Rn we have

d2.x;F .�// D max.a;˛/2C.�/

d2 .x;H .a; ˛// D max.a;˛/2C.�/


;

where H .a; ˛/ WD fx 2 Rn W a0x � ˛g. The proof has different steps:

(1) d2 .x;H .a; ˛// D Œ˛�a0x�Ckak2 is the well-known Ascoli formula.

(2) The inequality “�", i.e., d2.x;F .�// � sup.a;˛/2C.�/ d2 .x;H .a; ˛// followsfrom the fact that F .�/ � H .u; v/ for every .u; v/ 2 C .�/ :

(3) The converse inequality “�" follows from the following argument. In thenontrivial case, i.e., x … F .�/, consider its orthogonal projection Ox 2 F .�/.Then, u WD .x � Ox/ = kx � Oxk2 satisfies

d2.x;F .�// D kx � Oxk2 D u0 .x � Ox/ ;


and

u0 .z � Ox/ � 0 for all z 2 F .�/ :

(4) Since the constraint system of � is Farkas–Minkowski, by Farkas Lemma therewill exist � 2 R

.T /C such that

u

u0 Ox

!

DX

t2T�t

at

bt C bt

!

:

Then, defining � WDX

t2T�t (> 0, since u ¤ 0n), and

a

˛

!

WD ��1

u

u0 Ox

!

DX

t2T

�t

�

at

bt C bt

!

2 C .�/ ;

one has

d2.x;F .�// D u0 . Ox � x/ D ˛ � .a/0 xkak2

� max.a;˛/2C.�/


:

The following theorem characterizes the pseudo-Lipschitz property of F , undercontinuous perturbations of b, and gives an explicit formula for the Lipschitzmodulus.

Theorem 6.2.3 (Pseudo-Lipschitz Property of F , Canonical Perturbations).Let � D .c; a; b/ be a continuous LSIO and consider only continuous perturbationsof b (c and a remain fixed). Then, the following statements hold:

(i) F is pseudo-Lipschitz at .�; x/ for every x 2 F.�/ if and only if � satisfiesSCQ.

(ii) Assume that F is pseudo-Lipschitz at .�; x/. Then:(ii-1) If x is a Slater point for � , then lipF .�; x/ D 0.(ii-2) If x is not a Slater point for � , then

lipF .�; x/ D max

(1

kak2W a

a0x

!

2 C.�/)

> 0: (6.15)

Thanks to statement (i) and Theorem 5.1.1, together with the compactness ofC.�/, F is pseudo-Lipschitz at .�; x/ for every x 2 F.�/ if and only if 0nC1 …C.�/, and this happens if and only if F is lsc at �:


If we consider the marginal function associated with the nominal problem � ,g.x/ WD supt2T fbt � a0

t xg, it turns out that g.x/ > 0 if and only if x … F .�/. If.�; x/ 2 gphF with g.x/ D 0, we know that

@g.x/ D convn�at W bt � a0

t x D 0; t 2 ToD conv f�at W t 2 T .x/g ;

and if F is pseudo-Lipschitz at .�; x/ 2 gphF with g.x/ D 0, (6.15) can bewritten as

lipF .�; x/ D fd2.0n; @g.x//g�1 :

Example 6.2.2. We revisit again Example 1.1.1, and the three (optimal) pointsanalyzed there:

(a) If x1 D 02, then˚a 2 R

2 W �a; a0x1� D .a; 0/ 2 C.�/� D convf.1; 0/; .0; 1/g,

and lipF ��; x1� D p2.

(b) With x2 D .1=p2/ .1; 1/ we get fa 2 R2 W �a; a0x2

� D�a; .1=

p2/.a1 C a2/

�

2 C.�/g D f�.1=p2/ .1; 1/g, so that lipF ��; x2� D 1.(c) Finally, with x3 D .0; 1/,

˚a 2 R

2 W �a; a0x3� D .a1; a2; a2/ 2 C.�/

� Df.0;�1/g and lipF ��; x3� D 1.

A remarkably more involved case arises when perturbations of all the coefficientsare allowed. The following theorem deals with this case.

Theorem 6.2.4 (Pseudo-Lipschitz Property of F , General Perturbations). Let� D .c; a; b/ be an ordinary LSIO, and consider arbitrary perturbations of a andb. Then, the following statements hold:

(i) F is pseudo-Lipschitz at .�; x/ for every x 2 F.�/ if and only if � satisfiesthe SSCQ (or it satisfies any other condition equivalent to the lsc of F at �).

(ii) Assume that F is pseudo-Lipschitz at .�; x/ and that the set fat , t 2 T g isbounded, then :

(a) If x is a SS point for � , then lipF .�; x/ D 0.(b) If x is not a SS point for � , then

lipF .�; x/ D k.x;�1/k2 max

(1

kak2W a

a0x

!

2 clC.�/

)

> 0:

(6.16)

Again Theorem 5.1.1 yields the following equivalence: F is pseudo-Lipschitz at.�; x/ for every x 2 F.�/ if and only if 0nC1 … clC.�/, if and only if F is lsc at � .

The maximum in (6.16) is attained and it is positive. Observe that gphF isnot convex and, therefore, standard tools in variational analysis as the Robinson–Ursescu theorem (see, for instance, [75]) do not apply here.


6.2.3 Calmness of F

Let � D .c; a; b/ be a LSIO and consider again only perturbations of b (c and aremain fixed). The same function g can also be used for analyzing the calmness ofF at .�; x/ 2 gphF with g.x/ D 0. The calmness of F at .�; x/ 2 gphF reads asthe existence of � 0 and U 2 Nx such that

d2 .x;F.�// � d1��;F�1 .x/

�; for all x 2 U;

where

d1��;F�1 .x/

� D supnhbt � a0

t xi

C ; t 2 To

(6.17)

Dhsupfbt � a0

t x; t 2 T gi

CD Œg.x/�C:

Next, according to [165, Theorem 1] (and also to [9, Proposition 2.1 and Theorem5.1]), F is calm at .�; x/ 2 gphF with g.x/ D 0 if and only if

lim infx!x; g.x/>0

d2.0n; @g.x// > 0 (6.18)

and

clmF.�; x/ D�

lim infx!x; g.x/>0

d2.0n; @g.x//

��1: (6.19)

Example 6.2.3. Remember that in Example 1.1.1, for x1 D .0; 0/ we hadlipF ��; x1� D p2. Taking the sequence zr D .�1=r;�1=r/, r D 1; 2; : : :, whichconverges to x1, we observe g.zr / D 1=r and @g.zr / D convf�.1; 0/;�.0; 1/g.By (6.19)

clmF.�; x1/ �n

limr!1 d2.0n; @g.z

r //o�1 D p2 D lipF.b; x1/:

Since clmF.�; x1/ � lipF.�; x1/, we conclude that clmF.�; x1/ D p2.Now, for x3 D .0; 1/ we had lipF ��; x3� D 1, and if yr D .�1=r; 1 � 1=r/,

r D 2; 3; : : :, which obviously converges to x3, we obtain g.yr/ D 1=r , @g.yr/ Df�.1; 0/g. This time (6.19) yields

clmF.�; x3/ �n

limr!1 d2.0n; @g.y

r//o�1 D 1 D lipF.�; x3/;

and so clmF.�; x3/ D 1.


Remark 6.2.1 (Sources and Extensions).Distance to Ill-Posedness with Respect to Consistency: The equivalence between(i) and (iii) in Theorem 6.2.1 is Proposition 1 in [49], but the proof given hereof the equivalence between (ii) and (iii) comes from [127, Proposition 1]; and theequivalence with (ii) is established in [56, Theorem 3]. The implication (6.12)is given in [52, Proposition 3.2] without appealing to the marginal function g.Theorem 6.2.2 is [49, Theorem 6].

Pseudo-Lipschitz Property of the Feasible Set: Theorem 6.2.3 can be found in [34,Theorem 2.1 and Corollary 3.2], and Theorem 6.2.4 is [37, Theorem 1]. In [156] therelationships among metric regularity, metric regularity with respect to right-handside perturbations, and the extended Mangasarian–Fromovitz CQ are established ina non-convex differentiable framework. In [140] a local error bound is provided,under the boundedness of the set C .�/ but not requiring the existence of a strongSlater point.

Remark 6.2.2 (Alternative Characterizations of Calmness). Again we consideronly perturbations of b in � D .c; a; b/.(a) The following condition also characterizes the calmness of F at .�; x/ 2 gphF

with g.x/ D 0, and it constitutes a linear version of Theorem 3 in [159]: Thereexists �0 > 0 and a neighborhood U 2 Nx such that, for each x 2 U withg .x/ > 0 we can find associated ux 2 R

n, kuxk2 D 1, and �x > 0 satisfying

g.x C �xux/ � g .x/ � �x�0, for all t 2 T: (6.20)

(b) Let us recall here the well-known Abadie constraint qualification (ACQ forshort) at .�; x/, which in our framework may be written as

clD.F.�/I x/ D A .x/ı : (6.21)

Observe that the inclusion clD.F.�/I x/ � A .x/ı always holds (see (1.4)),and that clD.F.�/I x/ is the cone of tangents to F.b/ at x. Equation (6.21)yields to a kind of asymptotic optimality condition:

x 2 S.�/, c 2 .clD.F.�/I x//ı D A .x/ıı D clA .x/ :

An example, in R3, where ACQ fails at .�; x/ is given by

�.�/ WD� � .cos t / x2 � .sin t / x3 � bt ; t 2 Œ0; �=2� ;

x3 � bt ; t D 2;�

where T D Œ0; �=2�[ f2g, bt D �1 if t 2 Œ0; �=2�, b2 D 1, and x D .0; 0; 1/0.In this case F.�/ D R��1; 0��f1g, and clD.F.�/I x/ D R��1; 0��f0gwhereas A .x/ı D fa�=2; a2gı D R

2 � f0g.


In [57, Theorem 3] it is proved that the fulfilment of ACQ at .�; x/ forx 2 F.�/ \ U , where U is a neighborhood of x, together with an additionalproperty of uniform boundedness of the scalars generating the unit vectors inA .x/, constitute conjointly a characterization for calmness of F at .�; x/ 2gphF with g.x/ D 0. This characterization is strongly based on [236, Theorem2.2]. Section 4 in [57] provides an operative formula for computing clmF.�; x/in the case when T is finite.

Remark 6.2.3 (Global Error Bound). In [55, Proposition 4.5] the following globalerror bound is obtained: Given � 2 domF , suppose that there exist K, Ox, and" > 0 such that kxk2 � K for all x 2 F.�/ and a0

t Ox � bt C " for all t 2 T .Then "�1.1 C k Oxk2/maxf1;Kg is a global error bound for � . For Example 1.1.1,Ox D .1=2; 1=2/ is a Slater point with associated " D 1 � 1=p2, and kxk2 � 1 forall x 2 F.�/. Hence, 2

p2 C 3 is the global error bound given by the expression

above.

Remark 6.2.4 (Radius of Metric Regularity). Since the pseudo-Lipschitz propertyof F at .�; x/ is equivalent to the metric regularity of F�1 at .x; �/, it makes senseto consider the notion of radius of metric regularity, which is defined as follows:

radF�1 .x; �/ WD inf`2L

�k`k W F

�1 C ` is not metricallyregular at .x; �/C ` .x//

�;

where L is the space of linear continuous mappings from Rn into C .T;R/, i.e.,

`.x/ D '.�/0x for a certain ' 2 C .T;Rn/. Consequently, if we consider thesupremum norm in C .T;R/, one gets

k`k D supkxk�1

k`.x/k1 D supkxk�1

��'.�/0x��1

D maxt2T sup

kxk�1

ˇ'.t/0x

ˇ D maxt2T k'.t/k� :

Additionally,

.F�1 C `/.x/ D .aC '.�//0x � CC .T;R/ :

The following result, called radius theorem, is established in [34, Theorem 3.1](other radius theorems are given in [74, 144, 189]): If .�; x/ 2 gphF , with � D.c; a; b/ continuous, allowing only continuous perturbations of b, then:

radF�1 .x; �/ D 1

lipF .�; x/ :

Remark 6.2.5 (Pseudo-Lipschitz Property of the Boundary Mapping). Larriquetaand Vera de Serio [166] analyze the equivalence between the pseudo-Lipschitz


Fig. 6.6 Distance to ill-posedness

property of the boundary set mapping, B, and the stability of the feasible setmapping F with respect to the consistency. The paper also studies the relationshipbetween the regularity moduli of B�1 and F�1, and provides conditions to assurethat the metric regularity of B�1 is equivalent to the lower semi-continuity of B,whose many characterizations are described in Sect. 5.1.

6.3 Quantitative Stability of the Optimal Set Mapping

6.3.1 Distance to the Ill-Posedness with Respect to Solvability

First, in this section, we provide a formula for the distance to the ill-posednesswith respect to solvability. For the sake of simplicity, we shall deal only with theEuclidean distance.

Theorem 6.3.1 (Distance to Ill-Posedness w.r.t. Solvability). If � D .c; a; b/ 2cl domS , then

d2.�; bd domS/ D minfd2.0nC1; bdH .�//; d2.0n; bdZ� .�//g; (6.22)

where Z� .�/ WD convfat ; t 2 T I �cg.In Fig. 6.6, ext.H/ represents the exterior of H , i.e., ext.H/ D int.RnC1ŸH/.

In the example illustrated in this figure, d2.0n; bdZ� .�// < d2.0nC1; bdH .�//.For any possible problem � considered in Example 1.1.1 (i.e., for any possible c),

F.�/ is bounded as M .�/ D Rn. Then, F is uniformly bounded in some

6.3 Quantitative Stability of the Optimal Set Mapping 97

neighborhood of � according to Lemma 5.1.1. So, � 2 int domS � cl domS .Consequently, theorem above applies and we see that

d2.0n; bdZ� .�// D .2/�1=2 > .5C 2p2/�1=2 D d2.0nC1; bdH .�//;

and d2.�; bd domS/ D .5C 2p2/�1=2.Example 6.3.1. Consider the linear programming problem

� W infx2R2

x2 s.t. � x1 C x2 � 0; x1 C x2 � 0; .1=4/x2 � 0:

Since

03 … H.�/ D convf.�1; 1; 0/; .1; 1; 0/; .0; 1=4; 0/g C RCf.0; 0;�1/g

we have and � 2 int domF by Theorem 5.3.1(i), and moreover

02 2 intZ�.�/ D int convf.�1; 1/; .1; 1/; .0; 1=4/;�.0; 1/g;

entailing, by Theorem 5.3.1(ii), � 2 int domS . Now,

1=p5 D k.2=5;�1=5/k2 D d2.02; bdZ� .�//

> d2.03; bdH .�// D k.0; 1=4; 0/k2 D 1=4;

and

d2.�1; bd domS/ D 1=4:

A problem Q� 2 bd domS such that d2.�; bd domS/ D d2.�; Q�/ is the following

Q� W infx2R2

x2 s.t.

8<

:

.�1 � 0/x1 C .1 � 1=4/x2 � 0 � 0;

.1 � 0/x1 C .1 � 1=4/x2 � 0 � 0;

.0 � 0/x1 C .1=4 � 1=4/x2 � 0 � 0:

In fact, Q� 2 bd domF as 03 2 bdH . Q�/.

6.3.2 Pseudo-Lipschitz Property of S

From now on we approach the study of S only in the context of continuousparameters � D .c; a; b/, with fixed left-hand side a, and continuous perturbationsof c and b. Let us start by considering the following example:


Fig. 6.7 S is not pseudo-Lipschitz

� W Infx2R2

x1 s.t. x1 � x2 � 0; x1 C x2 � 0; x1 � 0: (6.23)

Here � D .c; a; b/, with Nc D .1; 0/ and Nb D .0; 0; 0/. Obviously, S.�/ D f.0; 0/g.In order to see that S is not pseudo-Lipschitz at .�; x/, consider the perturbedproblems (we omit the fixed parameter a):

�r ��1;�1=r2� ; .0; 0; 1=r/� ; Q�r

��1; 1=r2

�; .0; 0; 1=r/

�; r D 1; 2; : : :

It is straightforward that

S .�r/ D f.1=r; 1=r/g ; S . Q�r/ D f.1=r;�1=r/g ; r D 1; 2; : : : :

Therefore

d2 .S .�r/ ;S . Q�r// D 2=r; while d1 .�r ; Q�r/ D 2=r2; r D 1; 2; : : : ;

and

lip S .�; x/ D lim sup.x;�/!.x;�/

d2 .x;S .�//d1 .�; .S/�1 .x// � lim

r!12=r

2=r2D1:

See also Fig. 6.7.In contrast with the situation in the previous example, we show next a problem

in which we cannot conclude (see Fig. 6.8) by using the strategy used above, that Sdoes not enjoy the pseudo-Lipschitz property. The new problem is

Q� W Infx2R2fx1 C .1=2/x2g s.t. x1 � x2 � 0; x1 C x2 � 0; x1 � 0; (6.24)

for which we have again S . Q�/ D f.0; 0/g.


Fig. 6.8 Perturbations do not preclude pseudo-Lipschitz

The following notion is crucial in our approach: given .�; x/ 2 gphS , theextended Nürnberger condition (ENC, in brief) is held at .�; x/ if � satisfies theSCQ, and there is no E � T .x/ with jEj < n such that c 2 cone fat ; t 2 Eg.

The following result gives a complete characterization of the pseudo-Lipschitzproperty of S for the LSIO problem in our continuous context. It shows theequivalence among ENC and: (1) the pseudo-Lipschitz property of S at .�; x/,(2) the strong Lipschitz stability of S at .�; x/ or strong metric regularity of S�1at .x; �/ [75, 3G], and (3) a Kojima’s type stability under specific perturbations[160]. The fact that the pseudo-Lipschitz property of the optimal set mapping of aparametric optimization problem implies strong Lipschitz stability holds for a rathergeneral class of optimization problems (see, e.g., [158, Corollary 4.7], [75, Theorem4F.7]).

Theorem 6.3.2 (Pseudo-Lipschitz Property of S for Continuous LSIO’s). Letus consider the continuous LSIO problem � D .c; a; b/ with continuous per-turbations of Nc and Nb, and .�; x/ 2 gphS . Then, the following conditions areequivalent:

(i) S is pseudo-Lipschitz at .�; x/.(ii) S is single-valued and Lipschitz continuous in some neighborhood of � .

(iii) S is single-valued and continuous.(iv) S is single valued in some neighborhood of � .(v) ENC holds at .�; x/.

Observe that ENC fails for the problem � in (6.23) but it is held for problem Q�in (6.24), entailing that S is pseudo-Lipschitz at . Q�; 02/. In Example 1.1.1, with bas it was defined there, S is pseudo-Lipschitz only at the following pairs:

..c1; c2/; b/; x/ for

8<

:

x D .1; 0/ and c1 < 0; c2 > 0;x D .0; 1/ and c1 > 0; c2 < 0;x D .0; 0/ and c1 > 0; c2 > 0:


ENC leads us to consider the nonempty family of sets

T.x/ WD fE � T .x/ W jEj D n and c 2 conefat ; t 2 Egg : (6.25)

For problem Q� in (6.24), T.02/ D ff1; 2g; f2; 3gg.It is obvious that, under ENC, if E 2 T.x/, the vectors fat , t 2 Eg are linearly

independent and so the matrix AE whose rows are these vectors is non-singular. In[38, Theorem 1] the authors proved that if S is pseudo-Lipschitz at .�; x/ 2 gphS ,then

lip S .�; x/ � supE2T. Nx/

��A�1E

�� :

Moreover, the equality holds when T is finite (which solves an open problemproposed in [174]). Here we identify AE with the endomorphism x 7! AEx, withthe images space being endowed with the supremum norm k�k1. Provided that AEis non-singular, we have

��A�1E

�� D maxkyk1�1

��A�1E y

��2D max

y2f�1;1gn��A�1

E y��2: (6.26)

The second equality comes from the fact that f�1; 1gn is the set of extreme pointsof the closed unit ball corresponding to the norm k�k1, plus the convexity of thefunction to be maximized.

For problem Q� in (6.24), lip S . Q�; 02/ D maxf��.a1; a2/�1��2;��.a2; a3/�1

��2g D

51=2.

6.3.3 Calmness of S

Again in the context of continuous parameters � D .c; a; b/, with continuousperturbations of c and b, calmness of S is analyzed through the calmness of thepartial solution set mapping Sc W C .T;R/ � R

n given by

Sc .b/ WD S .c; b/ ,

and the so-called (lower) level set mapping L W R � C .T;R/ � Rn defined by

L .˛; b/ WD ˚x 2 Rn j c0x � ˛I a0

t x � bt ; t 2 T�:

The calmness of L at ..c0x; b/; x/ 2 gphL reads as the existence of � 0 and aneighborhood U 2 Nx such that

d2

�x;L.c0x; b/

�� d1

�.c0x; b/;L�1 .x/

�; for all x 2 U;


where, obviously, L.c0x; b/ D S.�/, and

d1�.c0x; b/;L�1 .x/

�D supfc0x � c0x

C I

hbt � a0

t xi

C ; t 2 T g

Dhsupfc0x � c0xI bt � a0

t x; t 2 T gi

C : (6.27)

If we introduce the function h W Rn ! R

h .x/ WD supnc0x � c0xI bt � a0

t x; t 2 To

D max˚c0x � c0x; g.x/

�; (6.28)

it turns out that

L.c0x; b/ Dnx 2 R

n W h .x/ � 0o

and d1�.c0x; b/;L�1 .x/

�Dhh .x/

i

C :

Moreover

@h .x/ D8<

:

convn�at W bt � a0

t x D h .x/ ; t 2 To; if c0x � c0x < h .x/ ;

conv�n�at W bt � a0

t x D h .x/ ; t 2 To[ fcg

�; if c0x � c0x D h .x/ :

In particular

@h .x/ D conv .f�at W t 2 T .x/g [ fcg/ : (6.29)

As a is fixed, and for having a better formulation of the next statements, we shall

identify � with .c; b/. Then, if�.c; b/; x

�2 gphS satisfies SCQ, Proposition 6.3.1

below establishes the equivalence among the calmness of S at�.c; b/; x

�and the

calmness of L at�.c0x; b/; x

�, allowing us to use the characterization for calmness

of the set mapping given in (6.18):

Proposition 6.3.1 (Calmness of S). If�.c; b/; x

�2 gphS , and we assume that

SCQ holds, then the following statements are equivalent:

(i) S is calm at�.c; b/; x

�I

(ii) Sc is calm at .b; x/I(iii) L is calm at

�.c0x; b/; x

�I

(iv) lim infx!x; h.x/>0 d2.0n; @h .x// > 0.

Now we provide a theorem characterizing the isolated calmness of the argminmapping:


Theorem 6.3.3 (Isolated Calmness of S). Let � D .c; a; b/, with c ¤ 0n, be acontinuous LSIO and consider only continuous perturbations of Nc and Nb. If S.�/ Dfxg and SCQ holds, then the following assertions are equivalentW

(i) c 2 intA .x/ I(ii) x is a strongly unique minimizer of � I

(iii) S is isolatedly calm at ..c; b/; x/I(iv) Sc is isolatedly calm at .b; x/.

If T is finite, the conditions above are equivalent to the uniqueness of x as optimalsolution of � .

As the authors of [42] pointed out, the equivalence (i),(iii) can also be obtainedfrom Theorem 6.3.1. Thanks to (6.29), c 2 intA .x/ is equivalent to

0n 2 int .conv.f�at W t 2 T . Nx/g [ f Ncg// D int @h. Nx/:

Let > 0 be such that �1B2 � @h. Nx/. Then, for every x 2 R

n we have

hu; x � Nxi � h.x/ for all u 2 B2;

and, if x ¤ Nx, taking u D .x � Nx/= kx � Nxk2 we get

d2.x;S.c; b// D kx � Nxk2 � h.x/I

hence, taking into account that S.c; b/ D L.c0x; b/, Theorem 6.3.1 yields theaimed calmness property. The same argument (applied backwards) also shows thenecessity of c 2 intA .x/. We call the attention of the reader on the fact that wehave actually proved that (i) is equivalent to the existence of a global error bound ofh at x.

In the rest of this section we approach the problem of estimating clmS..c; b/; x/.More precisely, we shall provide a lower bound for this modulus under SCQ anduniqueness of the optimal solution. It turns out that this lower bound is also upperbound in linear programming, and therefore, it represents the exact modulus in linearprogramming under uniqueness of the optimal solution. In addition, we shall give anew upper bound which has the virtue of relying only on the nominal data.

Remember that we are considering all the time a continuous LSIO � D .c; a; b/,with continuous perturbations of Nc and Nb, and assume that SCQ holds. Associatedwith x 2 S.c; b/ we introduce the family of active index sets:

K .x/ WD fE � T .x/ W jEj � n and c 2 conefat ; t 2 Egg :

K .x/ ¤ ; by Theorem 1.1.3, and it relates to the different KKT representations ofvector c, after applying Carathéodory Lemma.


Fig. 6.9 Interpretation of thefunction fD

Associated with each E 2 K .x/ we consider the supremum function

fE .x/ WD supnbt � a0

t x; t 2 T I � .bt � a0t x/; t 2 E

o

D supnbt � a0

t x; t 2 TŸEIˇˇa0t x � bt

ˇˇ; t 2 E

o:

Obviously fE .x/ � 0 for all x 2 Rn, and fE .x/ is the smallest perturbation size

on b to make x be a KKT point (hence optimal), with E as an associated KKT set.Let us illustrate the meaning of the function fE with a simple example:

� W Infx2R2fx1 s.t. x1 � 0; x1 � .1=4/x2 � 0; x2 � 0g:

Here b D 03 and the only optimal solution is x D 02. For E D f1g and x D .1; 6/

we see in Fig. 6.9 that

ff1g .1; 6/ WD supf�1;�1C .1=4/6;�6;�.�1/g D 1:

Only constraints 1 and 2 need to be perturbed in order to get optimality at x withc 2 cone fa1g (and the perturbation size is ff1g .x/ D 1).

Let us give some further information about the functions fE , E 2 K .x/ W1. fE W Rn ! RC is a convex function.2. ŒfE D 0� � S.c; b/ for all E 2 K .x/.3. Under uniqueness, i.e., if S.c; b/ D fxg, we have ŒfE D 0� D S.c; b/ for allE 2 K .x/, and one can also check that, for x … S.c; b/,

d2.x; x/ d2.x;S.c; b// � fE .x/

d2.0n; @fE .x//:

The following result is established in [44, Theorem 5].


Theorem 6.3.4 (Lower Bound for the Calmness Modulus of S). Consider thecontinuous LSIO � D .c; a; b/, with continuous perturbations of Nc and Nb. If weassume that SCQ holds and that S.c; b/ D fxg, then

clmS..c; b/; x/ � clmSc.b; x/ � supE2K.x/

lim supx!x

fE.x/>0

1

d2 .0n; @fE .x//:

We consider next the important particular case of linear programming. Thefollowing theorem [44, Theorem 6 and Corollary 1] shows that the lower boundabove for clmS �� Nc; Nb� ; x� is also an upper bound, without requiring neither SCQnor the uniqueness of x as an optimal solution of � .

Theorem 6.3.5 (1st Upper Bound for the Calmness Modulus of S in LO).Assume that T is finite. If x 2 S.c; b/, then

clmS..c; b/; x/ D clmSc.b; x/ � supE2K.x/

lim supx!x

fE.x/>0

1

d2 .0n; @fE .x//:

If, in addition, S.c; b/ D fxg, the inequality above becomes an equality.

The following step is addressed to provide an upper bound for the calmnessmodulus in linear programming relaying only on problem’s data. Recall that inlinear programming, according to Theorem 6.3.3,

S.c; b/ D fxg , c 2 intA .x/ ;

which entails the existence of some subset of indices E � T .x/, with jEj D n,such that

c 2 conefat ; t 2 Eg and AE WD .at /t2E is non-singular.

In other words, the following family of active index sets is nonempty:

G.x/ WD fE 2 T.x/ W AE non-singularg ;

where the family T.x/ has been introduced in (6.25). In fact, under the ENCcondition, and thanks to Carathéodory Lemma, both families coincide, i.e., G.x/ DT.x/. In [44, Theorem 8] the following upper bound is given:

Theorem 6.3.6 (2nd Upper Bound for the Calmness Modulus of S in LO). IfT is finite and Nc 2 intA.x/, then

clmS..c; b/; x/ � maxE2G.x/

��A�1E

�� : (6.30)


Table 6.1 Calmness moduliof S in LO and LSIO LSIO under SCQ LO

Not requiringuniqueness

Open problemclmSc.b; x/D clmS..c; b/; x/� C1

S.c; b/ D fxg C1 � clmSc.b; x/� clmS..c; b/; x/

clmSc.b; x/D clmS..c; b/; x/D C1 � C2

Fig. 6.10 Second upperbound attained (a)

Table 6.1, where

C1 WD supE2K.x/

lim supx!x

fE.x/>0

1

d2 .0; @fE .x//and C2 WD max

E2G.x/��A�1

E

�� ;

summarizes the results in this subsection (Fig. 6.10).

Example 6.3.2 (2nd Upper Bound Attained). Consider the problem

� W Infx2R2 fx1 C .1=3/x2 j x1 � 0; x1 C .1=2/x2 � 0; x1 C x2 � 0g :

We easily verify that G .x/ D ff1; 2g ; f1; 3gg,��A�1

f1;2g�� D p17, and

��A�1

f1;3g�� D p5

(according to (6.26). If we consider now the sequence ..br ; xr //1nD1 � gphS Nc givenby br D .1=r;�1=r; 0/ and xr D .�1=r; 4=r/, we have (see also Fig. 6.11)

limr!C1

kxr � Nxk2��br � Nb��1D limr!C1

�� 1r; 4r

� � .0; 0/��2�

�� 1r; 1r; 0� � .0; 0; 0/��1

D p17:

This expression, together with Theorem 6.3.6, entails clmS �� Nc; Nb� ; x� D p17.


Fig. 6.11 Second upper bound attained (b)

Example 6.3.3 (2nd Upper Bound Not Attained). Consider the problem

� W Infx2R2fx1 C .1=3/x2 s.t. x1 � 0; x1 C .1=2/x2 � 0;x1 C x2 � 0; x1 � x2 � 0g:

Now G .x/ D ff1; 2g ; f1; 3g ; f2; 4g ; f3; 4gg and so

��A�1f1;2g

�� Dp17;

��A�1f1;3g

�� Dp5;��A�1

f2;4g�� Dp17=3;

��A�1f3;4g

�� D 1:

On the other hand, one can check via a non-simple direct calculus (see [44]) that

clmS�.c; b/; x

�D p5 <

��A�1

f1;2g�� :

Remark 6.3.1 (Sources and Related Results).Distance to the Ill-Posedness with Respect to Solvability: The results in thissubsection, in particular Theorem 6.3.1, are in [50, Theorems 2, 5 and 6].

Pseudo-Lipschitz Property of the Optimal Solution: Theorem 6.3.2 is [43, Theorem16].

Calmness of the Optimal Solution: Theorem 6.3.1 can be found in [42, Theorem3.1]. Theorem 6.3.3, characterizing the isolated calmness of the argmin mapping, isTheorem 3 in [35]. The second upper bound in (6.30) equals the Lipschitz moduluswhen S is Aubin continuous, according to [37, Corollary 2].


In [39, 40] the following expression for lipS .�; x/ is established under ENC:

lipS�.c; b/; x

�D lipSc

�b; x

�D lim sup.z;b/!.x;b/

fb.z/>0

�d2.0n; [email protected]//

��1;

where

fb.z/ WD d1�b; .Sc/�1 .z/

�;

.Sc/�1 .z/ D fb 2 C.T;R/ W z 2 S.c; b/g, and [email protected]/ represents the Fréchetsubdifferential of fb at z: Moreover, ENC also allows us to represent fb in termsof certain difference of convex (d.c.) functions, which rely directly on the nominaldata [41].

Calmness of S at ..c; b/; x/ always holds when T is finite, even withoutassuming SCQ at b. The result follows from [204] (see also [207, Example 9.57] or[75, Theorem 3D.1]) as a consequence of the piecewise polyhedrality of gphS (i.e.,the graph is the finite union of polyhedral sets). Observe that gphL is polyhedral inthe finite case, so that the calmness of S at ..c; b/; x/ can be alternatively derivedfrom Theorem 6.3.1 together with the aforementioned result of Robinson, evenwithout SCQ, since KKT conditions hold for finitely constrained linear problemswithout SCQ.

Remark 6.3.2 (Two Open Problems in Quantitative Stability of LSIO Problems).

1. Efficient numerical methods for the computation of distances from a givenproblem to ill-posedness in some of the senses discussed in Sects. 6.2.1 and 6.3.1.

2. Efficient numerical methods for the computation of the Lipschitz and calmnessmoduli of F and S , specially for finite T .

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http://arxiv.org/abs/1301.0810

Index

Aassumption

density, 15Aubin continuity, 79

Bbound

exact calmness, 82global error, 85global error condition rate, 85local error, 82

Ccenter

analytic, 16geometric, 16

conditioncomplementarity, 7extended Nürnberger (ENC), 99Karush-Kuhn-Tucker (KKT), 6

coneactive, 5characteristic, 4first moment, 4of feasible directions, 5pointed, 3positive polar, 3recession, 3robust moment, 42second moment, 4

constraint qualificationFarkas-Minkowsky (FMCQ), 5local Farkas-Minkowsky (LFMCQ), 5Slater (SCQ), 5strong Slater (SSCQ), 5

constraintsdata set, 4Value-at-Risk, 34

Ddistance to ill-posedness, 30duality

robust, 42

Ffunction

affine, 53affinity cone of a, 53affinity partition of a, 53barrier, 16characteristic, 25concave, 25convex, 2directional derivative of a, 52domain of a concave, 25domain of a convex, 2dual optimal value, 28epigraph of a, 2graph of a, 2hypograph of a, 2Lagrange, 6logarithmic barrier, 16lower semicontinuous (lsc), 24lsc hull of a, 24marginal, 2membership, 25positively homogeneous, 28primal optimal value, 28proper, 2quasiconcave, 25quasiconvex, 2

M.A. Goberna and M.A. López, Post-Optimal Analysis in Linear Semi-InfiniteOptimization, SpringerBriefs in Optimization, DOI 10.1007/978-1-4899-8044-1,© Miguel A. Goberna, Marco A. López 2014

119

120 Index

function (cont.)ranking, 25Roubence ranking, 25slack, 5subdifferential of a concave, 25subdifferential of a convex, 2superlinear, 53support, 36upper semicontinuous (usc), 24usc hull of a, 24

Ggeneralized finite sequence, 3

Hhull

affine, 3convex, 3convex conical, 3linear, 3

Iinequality

consequent, 4trivial, 6weak dual, 10

Llemma

non-homogeneous Farkas, 4uniform boundedness of the feasible set, 63

limitinner, 61outer, 61

linearity of a convex cone, 3Lipschitz continuous around a point, 81Lipschitz-like, 79

Mmapping

(lower) level set, 100boundary, 65calmn, 81closed, 28continuous, 61dimensionally stable, 62domain of a, 29dual feasible set, 28dual optimal set, 28

extreme points set, 65graph of a, 29Hausdorff lower semicontinuous (H-lsc),

70Hausdorff upper semicontinuous (H-usc),

70immobile w.r.t. the affine hull, 62inner semicontinuous (isc), 61isolated calmn, 82locally upper Lipschitz, 82lower semicontinuous (lsc), 28metric projection, 70outer semicontinuous (osc), 61partial solution set, 100primal feasible set, 28primal optimal set, 28projection, 4stable in Tuy’s sense, 62topologically stable, 62uncertain set-valued, 40upper semicontinuous (usc), 28

metric regularity, 79modulus

calmness, 82

Nnumber

fuzzy, 25

Ooptimal set

dual, 10primal, 2

optimal valuedual, 10primal, 2

Ppartition

dual, 74maximal, 12maximal optimal, 12optimal, 12primal, 74

pointsaddle, 6Slater, 5strong Slater, 5

problemcontinuous, 1continuous dual, 11Haar’s dual, 10

Index 121

ill-posed, 74Lagrangian dual, 10non-degenerate, 65optimistic counterpart, 40robust counterpart of a, 39robust dual (optimistic counterpart), 40sublevel, 14totally ill-posed, 75well-posed, 74

pseudo-Lipschitz, 79Lipschitz modulus, 80

Rradius

consisteny, 43metric regularity, 95

Sscenario, 24selection, 40set

cardinality of a, 15dimension of a convex, 3dual feasible, 10dual support, 12fuzzy, 25hypographical, 43of active constraints, 5primal feasible, 1primal support, 12relative boundary of a, 3relative interior of a, 3strictly convex, 56sublevel, 2universal, 25

solutioncomplementary, 11strongly unique, 7

space of parameters, 27strictly continuous at a point, 81subspace

orthogonal, 3system

constraint, 4equivalent, 4inconsistent, 4reinforced, 64semi-infinite, 29strongly inconsistent, 4

Ttheorem

abnormal systems, 87

Affinity along segments, 55affinity along segments, 57affinity on neighborhoods, 53, 57affinity on polytopes, 54, 58affinity partition of the optimal value

function, 53attainability of the consistency radius, 44calculus of the consistency radius, 43calmness of the optimal set, 101converse strong duality, 11dimensional stability of the restricted

feasible set, 73Distance to ill-posedness w.r.t. consistency,

88distance to ill-posedness w.r.t. solvability,

96estimation of the optimal value, 52existence, 4first lower bound for the calmness modulus

of the optimal set in LO, 104global robust duality, 42isolated calmness of the optimal set, 102local robust duality, 42Lower bound for the calmness modulus of

the optimal set, 104lower semicontinuity of the feasible set, 62Lower semicontinuity of the restricted

feasible set, 73optimality, 7primal well-posedness, 75primal-dual well-posedness, 76Pseudo-Lipschitz property (canonical

perturbations), 91Pseudo-Lipschitz property (general

perturbations), 92Pseudo-Lipschitz property of the optimal

set, 99quadratic behavior on polytopes, 59radius, 95second lower bound for the calmness

modulus of the optimal set in LO,104

stability of the optimal set, 66stability of the optimal value, 66strong duality, 11total ill-posedness, 75tractability of the robust counterpart, 45uniqueness, 7Upper semicontinuity of of the feasible set

and the reinforced system, 65Upper semicontinuity of the feasible set, 63

VVoronoi cell, 70

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Post-Optimal Analysis in Linear Semi-Infinite Optimization