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Books in the Classics in Applied Mathematics series are monographs and textbooks declared out of printby their original publishers, though they are of continued importance and interest to the mathematicalcommunity. SIAM publishes this series to ensure that the information presented in these texts is not lost totoday's students and researchers.Editor-in-ChiefRobert E. O'Malley, Jr., University of WashingtonEditorial BoardJohn Boyd, University of Michiganeter Hoff, University of WashingtonLeah Edelstein-Keshet, University of Britishark Kot, University of WashingtonColumbiaeter Olver, University of MinnesotaW illiam G . Faris, University of Arizonahilip Protter, Cornell UniversityNicholas J. Higham, University of Manchestererhard Wanner, L'Universite de GeneveClassics in Applied M athema ticsC. C. Lin and L. A. Segel, M athematics A pplied to D eterministic Problems in the Natural SciencesJohan G. F. Belinfante and Bernard Kolman, A Survey of L ie Groups and Lie Algebras withApplications an d Computational MethodsJames M . Ortega, Num erical Analysis: A Second CourseAnthony V. Fiacco and Garth P. McCormick, Nonl inear Program ming: Seq uential Unconstrained MinimizationTechniquesF. H. Clarke, Optimization and N onsm ooth AnalysisGeorge F. Carrier and Carl E. Pearson, Ordinary Differential EquationsLeo Breiman, ProbabilityR. Bellman and G. M . Wing, An Introduction to Invariant ImbeddingAbraham Berman and Robert J. Plemmons, Nonnegative Matrices in the M athematical SciencesOlvi L. Mangasarian, N onlinear Programm ingCarl Friedrich Gauss, Theory of the Com bination of Obse rvations L east Sub ject to Errors: Part One, Part T w o,Supplement . Translated by G. W. Stewart (First tim e in p rint.)Richard Bellman, Introduction to Matrix AnalysisU. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution o f Boundary Value Problems f orOrdinary Diff erential EquationsK. E. Brenan, S. L. C ampb ell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in D ifferential-Algebraic EquationsCharles L. Lawson and Richard J. Hanson, Solving Least Squares ProblemsJ. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for Unconstrained Optimization and NonlinearEquationsRichard E. Barlow and Frank Proschan, Mathematical Theory of R eliabilityC ornelius Lanczos, L inear Differential OperatorsRichard Bellman, Introduction to M atrix Analysis, Second EditionBeresford N. Parlett, T he Symmetric Eigenvalue ProblemRichard Haberman, M athematical M odels: M echanical V ibrations, Population Dy nam ics, and Traffic FlowPeter W. M . John, S tatistical Design and Analysis of Exper imentsTamer Bazar and Geert Jan Olsder, D ynamic Noncooperative Game Theory, Second EditionEmanuel Parzen, Stochastic ProcessesPetar Kok otovic, Hassan K. Khalil, and John O'Reilly, Singular Perturbation M ethods in Con trol: Analysis andDesign


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Classics in Applied M athema tics (continued)Jean D ickinson Gibbons, Ingram O lkin, and M ilton Sobel, Selecting and Ordering P opulations: A NewStatistical MethodologyJames A. M urdock, Perturbations: Theory and MethodsIvar Ekeland and Roger Temam, C onvex Analysis and V ariational ProblemsIvar Stakgold, Boundary V alue Problems of Mathemat ical Physics, Volumes I and IIJ. M. Ortega and W . C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several VariablesD avid Kinderlehrer and Guido Stamp acchia, An Introduction to V ariational Inequalities an d The i rApplicationsF. Natterer, The M athem ati cs o f Com puteri zed TomographyAvinash C. Kak and M alcolm S laney, Principles of Computerized Tomographic ImagingR. Wong, Asymptotic Approximations of IntegralsO. Ax elsson and V. A. Barker, Finite Element Solution of Boundary Value Problems: Theory and C omputationDavid R. Brillinger, Time Series: D ata Analysis and TheoryJoel N. Franklin, M ethods of M athematical Economics: Linear and Nonlinear Programming, Fixed-Point TheoremsPhilip Hartman, Ordinary Differential Equations, Second EditionM ichael D . Intriligator, Mathematical Optimization and Economic TheoryPhilippe G. Ciarlet, T he Finite Element Method f or Elliptic ProblemsJane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large Symmetric Eigenvalue Computations,Vol. 1: TheoryM. Vidyasagar, Nonl inear Systems Analysis, Second EditionRobert Mattheij and Jaap Molenaar, Ordinary Differential Equations in Theory and PracticeShanti S. Gupta and S. Panchapakesan, Multiple Dec ision Procedures: The ory and Methodology of Selectingand Ranking PopulationsEugene L. Allgower and Kurt Georg, Introduction to Numerica l Continuation MethodsLeah Edelstein-Keshet, Mathematical Models in BiologyHeinz-Otto Kreiss and Jens Lorenz, Initial-Boundary Value P roblems and the N avier-Stokes EquationsJ. L. Hodges, Jr. and E. L. Lehmann, Basic Concepts of Probability and Statistics, Second EditionGeorge F. C arrier, M ax Krook, and C arl E. Pearson, Functions of a Complex Variable: Theory and TechniqueFriedrich Pukelsheim, Optimal Design of ExperimentsIsrael Gohberg, Peter Lancaster, and Leiba Rodman, Invariant Subspaces of M atrices with ApplicationsLee A. Segel with G. H. Handelman, M athem ati cs Applied to Continuum MechanicsRajendra Bhatia, Perturbation Bounds for M atrix EigenvaluesBarry C . Arnold, N. Balakrishnan, and H. N. Nag araja, A First C ourse in Order StatisticsCharles A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output PropertiesStephen L. Campbell and Carl D. Meyer, Generalized Inverses o f L inear TransformationsAlexander M organ, Solving Polynomial Systems Using C ontinuation for Engineering and Scientific ProblemsI. Gohberg, P. La ncaster, and L. Rodman, Matrix P olynomialsGalen R. Shorack and Jon A. Wellner, Empirical Processes with App lications to StatisticsRichard W. Cottle, Jong-Shi Pang, and Richard E. Stone, T he L inear Com plem entari ty ProblemRabi N . Bhattacharya and Edw ard C . Waym ire, Stochastic Processes with App licationsRobert J. Adler, The Geometry of Random FieldsMordecai Avriel, Walter E. Diewert, Siegfried Schaible, and Israel Zang, Generalized ConcavityRabi N. Bhattacharya and R. Ranga Rao, Normal Approx imation and Asymptotic Expansions


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PGeneralized Concavityb i

Mordecai AvrielTechnion Israel Institute of Technology

Haifa, IsraelW alter E. Diewert

University of British ColumbiaVancouver, British Columbia, Canada

Siegfried SchaibleChung Yuan Christian UniversityChung-Li, Taiwan

Israel Za ngThe Acade mic Co llege of Tel-Aviv-Yaffo

Tel-Aviv-Yaffo, Israel

sianLSociety for Industrial and A pplied M athematicsPhiladelphia


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Copyright 2010 by the Society for Industrial and Applied MathematicsThis SIAM edition is an unabridged republication of the work first publishedby P lenum Press, 1988.10987654321All rights reserved. Printed in the United States of America. No part of this bookmay be reproduced, stored, or transmitted in any manner without the writtenpermission of the publisher. For information, write to the Society for Industrialand Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA19104-2688 USA .Library of Cong ress Cataloging-in-Pub lication DataGen eralized concav ity / Mord ecai Avriel ... [et al.].p. cm. -- (Classics in applied mathematics ; 63)

Originally published: New Y ork: Plenum P ress, 1988.Includes bibliographical references and index.ISBN 978-0-898718-96-61. Concave functions. I. Avriel, M.QA353.C64G46 2010515'.94--dc22

2010035259

S.I.2JTL is a registered trademark.


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To my wife, Hava,our children, Dorit and Y oram ,R on and Ayelet, Tal and Dror,

and our grandchildren,Or, Shir, Yael, Y u v al, Sara, and Gal.

M ordecai Avriel

To my w i f e , Virginia Diewert.Erwin Diewert

To my late wife, Ingrid,our children, S u e , John, and Rickie,

and our grandchildren,Hannah, Joel, Elise, and Cody .

Siegfried Schaible

To my w i f e , Miri,our children, Ben, S hira, and Avi,

and our grandchildren, D an and Orr.Israel Zang


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ContentsPreface to the Classics Edition . . . . . . . . . . . . . . . . . . . .iPrefacev1. Introduction2. Concavity 52.1. Basic Definitions

52.2. Single-Variable Concave Functions . . . . . . . . . . . . . . . 162.3. Concave Functions of Several Variables . . . . . . . . . . . . . 242.4. Extrema of Concave Functions . . . . . . . . . . . . . . . . . 372.5. Concave Mathematical Programs . . . . . . . . . . . . . . . . 41

3. Generalized Concavity . . . . . . . . . . . . . . . .53.1. Quasiconcave FunctionsDefinitions and General Properties. . 553.2. Differentiable Quasiconcave Functions. . . . . . . . . . . . 673.3. Strict Quasiconcavity. . . . . . . . . . . . . . . . . . . . 63.4. Semistrict Quasiconcavity. . . . . . . . . . . . . . . . . . 13.5. Pseudoconcave Functions. . . . . . . . . . . . . . . . . . 93.6. Generalized Concave Mathematical Programs. . . . . . . . . 944. Application of Generalized Con cavity to Economics.014.1. The Costunction.014.2. Duality between Cost and Production Functions.124.3. Generalized Concavity and Producer Theory.184.4. The Consumer's Utility Maximization Problem .254.5. Semistrict Quasiconcavity and Consumer Theory.304.6. Strict Quasiconcavity and Consumer Theory.314.7. Strong Pseudoconcavity and Consumer Theory

.334.8. Pseudoconcavity and Consumer Theory

.374.9. Profit Maximization and Comparative Statics Analysis.394.10. Concave Programs and Economics.41ix


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xontents5 . Special Functional Form s I: Composite Functions, Products, and Ra tios .535.1. Composite Functions. . . . . . . . . . . . . . . . . . .545.2. Products and Ratios. . . . . . . . . . . . . . . . . . .586. Special Functional Forms II: Quadratic Functions . . . . . . . . . .676.1. Main Characterization of Quasiconcave, Pseudoconcave, andStrictly Pseudoconcave Quadratic Functions . . . . . . . . . .686.2. Characterizations in Terms of an Augmented Hessian . . . . . .826.3. Characterizations in Terms of the Bordered Hessian. . . . .866.4. Strictly Quasiconcave and Strongly Pseudoconcave Quadratic

Functions. .. . . . . . .. .. . . . . . . .906.5. Quadratic Functions of Nonnegative Variables . . . . . . . . .927 . Fractional Programming. . . . . . . . . . . . . . . . . . . .077.1. Notation and Definitions. . . . . . . . . . . . . . . . .077.2. Applications. . . . . . . . . . . . . . . . . . . . . . .107.3. Concave Fractional Programs. . . . . . . . . . . . . . .147.4. Algorithms in Concave Fractional Programming. . . . . . .248. Concave T ransformable Functions . . . . . . . . . . . . . . . . .31

8.1. Concave Transformable Functions: Range Transformation . . . .328.2. Conditions for Concavifiability of Twice ContinuouslyDifferentiable (C 2 ) Functions. . . . . . . . . . . . . . .428.3. Concave Transformable Functions: Domain and RangeTransformations. . . . . . . . . . . . . . . . . . . . .819. Additional Generalizations of Concavity. . . . . . . . . . . .939.1. F-Concave Functions. . . . . . . . . . . . . . . . . . .949.2. Generalized Arcwise Connected Functions . . . . . . . . . . .04Supplementary B ibliography. . . . . . . . . . . . . . . . . . .21Author Index. . . . . . . . . . . . . . . . . . . . . . . . . .25Sub ject Index. . . . . . . . . . . . . . . . . . . . . . . . . .2 9


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Preface to the Classics EditionWith the republication of our book, first published in 1988, we would like torecall how it all started.M ordecai Avriel worked on nonlinear programming, w here convexity playsa major role. If the nonlinear prog ram is minimizing a convex function (or max-imizing a concave function) over a convex set, every local optimum is a glob alone. C onsequently, searching for a g lobal optimum of a convex nonlinear pro-gram is much easier than seeking the optimum in a g eneral nonlinear program.In other words, convexity is a sufficient condition for local minima to be aglobal one, but it is definitely not a necessary one. Therefore, Mordecai beganstudying functions that are mo re g eneral than convex and in special cases canbe reduced to convex ones. He p ublished his first results in 1972, in the paper"r-Convex Functions."

Siegfried Schaible was working at about the same time on various formsof g eneralized convexity in his doctoral dissertation in Germany. Sieg fried andM ordecai met at the Department of O perations Research of Stanford University.At about the same time Israel Zang started his doctoral research at theTechnion in Haifa, Israel, under the supervision of Mordecai. The disserta-tion dealt with convex transformable functions and programs. The conceptof r-convex functions is concerned with a particular range transformation ofthe function under discussion. Israel's work created the framework to handlegeneral rang e and/or dom ain transformations. In addition, this dissertation dis-cusses the characterization of functions whose local minima are glob al throughcontinuity properties of their level sets mapping s.

The nex t phase of collaboration started when M ordecai spent a sabb atical atthe University of British Columb ia (UBC ) in Vancouver in 1974-1 975, wherehe met Erwin Diewert, who was interested in generalized convexity from aneconom ist's viewpoint. It turns out that convex ity (and concavity) have impor-tant implications in economic theory. There are three important constrainedmaximization or minimization problems that play a fundamental role in eco-nomics: (i) the cost minimization problem of a producer; (ii) the expenditure

X I


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Preface to the Classics Edition

minimization problem of a consumer; and (iii) the profit maximization prob-lem of a producer. The first two optimization problems g enerate a cost functionand an ex pend iture function, respectively, and it turns out that these tw o func-tions must be concave functions in the input prices that the producer faces andthe commodity prices that the consumer faces. The third optimization problemgenerates a profit function, which m ust be a convex function in the prices thatthe producer faces. Thus the theory of convex and concave functions plays avery important role in economics. Another area of economics where general-ized concave func tions play an imp ortant role is concerned w ith the followingquestion: W hat type of g eneralized concavity is required to guarantee that thesatisfaction of first order necessary conditions for max imizing a differentiablefunction will also be sufficient? The answer is that pseudoconcavity w ill do thejob.

Israel Zang completed his doctoral studies in 1974, and following a one-year visit to the Center for Operations Research and Econometrics (CORE),he joined the Faculty of Management at Tel Aviv University and was subse-quently invited to UBC in Vancouver in 1980-1982, where he continued hiscooperation with Erwin. During this period the JET paper on nine kinds ofquasi concavity and concavity, written jointly with Mordecai and Erwin, wasfinalized, and first versions of chapters of this book were w ritten.

The four researchers decided to organize a conference on the topic of gener-alized convexity. It was decided to ask for the support of N ATO, and indeed theconference was held as a NATO Summ er School in V ancouver in Aug ust 1980.At the conference, the four researchers also decided on writing the first bookof its kind on the subject of generalized convexity. Erwin thoug ht that in orderto interest economists in the b ook it should be called Generalized Concavity. Ittook 8 years until the book was finally published in 1988.

Since the b ook's original publication, three of the coauthors have turned toother fields.

M ordecai took early retirement from Technion and became D irector of An-alytic Development at Bank Hapoalim in Israel. He is head of a multidisci-plinary group w orking on financial engineering.

Erw in turned to more traditional subjects in economics. Up until 1985, heworked in various areas associated with microeconomic theory such as dual-ity theory, the pure theory of international trade, the measurem ent of waste inan economy, and nonparametric producer and consumer theory. Duality the-ory does have c lose connections w ith generalized concavity; i.e., one looks fortypes of g eneralized concave p roduction functions such that the dual cost func-tion can provide a complete representation of the primal production function.


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Preface to the Classics Editionii iAfter 1985, Erwin focused on economic measurement issues. Over the pasttwo decades, Erwin has been working with international agencies such as theOE C D , the IM F, and the Wo rld Bank in writing m anuals that address some ofthe above measurement issues.Israel turned to higher education administration. After serving as Deanof the Recanati Faculty of Management at Tel Aviv University, he becam e V iceProvost of the university, retiring w hen his term w as comp leted. He is currentlythe President of the Academic Co llege of Tel-Aviv-Yaffo.

Siegfried has remained in the field of generalized convexity. In 1994, heorganized a session cluster for the International Sym posium of the M athemat-ical Programming Society (ISMP). He had chosen as the focus for the clusterthe topic of g eneralized convexity. It turned out to b e the largest cluster on thattopic up to that point, with well over 10 sessions.By that point in time, Siegfried was also aware that researchers in ap-plied mathematics, including optimization, usually worked as individuals be-hind closed doors. This seemed to him to b e less efficient than a more comm unity-based, team-oriented work style such as that found in the natural sciences. Infact, he had read several biograp hies and found positive examples of those sci-entists who had favored a team app roach. So he set out to start a collaborativeeffort in generalized convexity; thus, the international Working Group of Gen-eralized Convexity (WGGC) was born. More information on the group canbe found at w w w .genconv.org. S ince its inception the group has organized thetriennial symposia on g eneralized convexity.

In time the WGGC also started to explore the related area of generalizedmonotonicity. The m ost notable event of the group w as the pub lication of theHandbook of General ize d Conv ex i ty and Generalized M onotonici ty (Springer,2005). Users of the Handbook soon learned that this book, General iz ed Con-cavity, was needed to benefit fully from the Handbook. Since GeneralizedConcavi ty was by then out of print, finding copies to use as reference texts wasquite difficult. Therefore, the authors welcomed the efforts of SIAM to returnthe original monograph to print in the SIAM Classics in Applied Mathemat-ics series. Together with the Handbook and the proceedings from each GCMconference, the community is now w ell served with key references on the topic.

We thank Sara M urphy and the other SIAM staff who w orked on the bookfor the great support in preparing the S IAM C lassics edition of our book .M ordecai AvrielWalter E. D iewertSiegfried SchaibleIsrael Zang


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PrefaceConcavity plays a central role in mathematical economics, engineering,management science, and optimization theory. The reason is that concavityof functions is used as a hypothesis in most of the important theoremsconcerning extremum problems. In other words, concavity is usually asufficient condition for satisfying the underlying assumptions of thesetheorems, but concavity is definitely not a necessary condition. In fact, thereare large families of functions that are nonconcave and yet have propertiessimilar to those of concave functions. Such functions are called generalizedconcave functions, and this book is about the various generalizations ofconcavity, mainly in the context of economics and optimization.

Although hundreds of articles dealing with generalized concavity haveappeared in scientific journals, numerous textbooks have specific chapterson this subject, and scientific meetings devoted to generalized concavityhave been held and their proceedings published, this book is the first attemptto present generalized concavity in a unified framework. We have collectedresults dealing with this subject mainly from the economics and optimizationliterature, and we hope that the material presented here will be useful inapplications and will stimulate further research.The writing of this book constituted a unique experience for the authorsin international scientific cooperationcooperation that extended overmany years and at times spanned three continents. It was an extremelyfruitful and enjoyable experience, which we will never forget.We are indebted to our respective home universitiesthe Technion-Israel Institute of Technology, the University of British Columbia, theUniversity of Alberta, and Tel Aviv Universityfor including the otherauthors in their exchange programs and for the technical assistance wereceived. Thanks are also due to the Center for Operations Research andEconometrics, Universite Catholique de Louvain, for the hospitality exten-ded to one of the authors.

xv


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xvirefaceThe writing of this book was partially supported by the Fund for theAdvancement of Research at Technion, the Natural Sciences and Engineer-ing Research Council of Canada, the Deutsche Forschungsgemeinschaft(West Germany), the U.S. National Science Foundation, and the IsraelInstitute of Business Research at Tel Aviv University.

M ordecai AvrielHaifa, Israel

Walter E. DiewertVancouver, British Columbia, Canada

Siegfried SchaibleEdmonton, Alberta, CanadaIsrael ZangTel Aviv, Israel


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1

IntroductionIn this introductory chapter, we provide a brief and mostly nonmathematicaldescription of the contents of this book on generalized concavity. Formalmathematical definitions of the various types of concavity may be foundin subsequent chapters.The first question we must attempt to answer in this chapter is: whydo concave functions occupy such an important position in economics,engineering, management science, and applied optimization theory in gen-eral? A real-valued function of n variables defined over a convex subset ofEuclidean n-dimensional space is concave if (if and only if) the line segmentjoining any two points on the graph of the function lies on or below thegraphs; a set is convex iff, given any two points in the domain of definitionof the set, the line segment joining the two points also belongs to the set.Returning to the question raised above, we suggest that the importanceof concave functions perhaps rests on the following two properties: (i) alocal maximizer for a concave function is also a global maximizer, and (ii)the usual first-order necessary conditions for maximizing a differentiablefunction f of n variables over an open set [i.e., x* is a point such that thegradient vector off vanishes so that Vf(x*) = 0] are also sufficient to implythat x* globally maximizes f if f is a concave function defined over a convexset. Various generalizations of concavity (studied in Chapter 3) preserveproperties (i) and (ii), respectively. In Chapter 2, we also study two classesof functions that are more restrictive than the class of concave functions:strictly concave and strongly concave functions. Strictly concave functionshave the following useful property, which strengthens property (i) above:(iii) A local maximizer for a strictly concave function is also the uniqueglobal maximizer. A function is strictly concave if the line segment joiningany two distinct points on the graph of the function lies below the graphof the function (with the obvious exception of the end points of the line


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Chapter t

segment). A function f is s t rong ly concave if it is equal to the sum of aconcave function and a negative definite quadratic form, i.e., f(x) =g(x) ax Tx for every x belonging to the convex domain of definition set,where g is a concave function, a > 0 is a positive scalar, and x Tx = x;.Strongly co ncave functions also have a property (iii) ab ove, and, in addition,in a neighborhood of a local maximizer, a strongly concave twice con-tinuously differentiable function w ill have the curvature of a neg ative definitequadratic form. This property is useful in proving convergence of certainoptimization algorithms, and it is also useful in enabling one to provecomparative statics theorems in economics; see Section 4.9 in Chapter 4.We now describe the contents of each chapter.C hapter 2 deals w ith concave functions and the two classes of functionsthat are stronger than concavity, namely, strictly and strongly concavefunctions. The first three sections of Chapter 2 develop alternative charac-terizations of concave functions. In addition to the definition of a concavefunction, there are three additional very useful characterizations of con-cavity: (i) the hypograph of the function (the graph of the function andthe set in (n + 1)-dimensional space lying below the graph) is a convex set;(ii) the first-order Taylor series approximation to the function around anypoint in the domain of definition lies on or above the graph of the function(this characterization requires the existence of first-order partial derivativesof the function); (iii) the Hessian matrix of second-order partial derivativesof the function evaluated at each point in the domain of definition is anegative semidefinite matrix (this characterization requires the existence ofcontinuous second-order partial derivatives of the function).Section 2.3 of Chapter 2 also develops some composition rules forconcave functions; e.g., a nonnegative sum of concave functions is a concavefunction or the pointwise minimum of a family of concave functions is a

concave function, and so on. Additional composition rules are developedin Chapter 5. Section 2.3 also provides characterizations for strictly andstrongly concave functions.Section 2.4 derives the local-global maximizer properties of concavefunctions referred to earlier. As we stated before, these properties areprobably the main reason for the importance of the concavity concept inapplied optimization theory.Section 2.5 deals with another extremely important topic from theviewpoint of applications, namely, concave mathematical programmingproblems. A concave program is a constrained maximization problem, w here(i) the objective function being maximized is a concave function; (ii) thefunctions used to define equal to or greater than zero inequality constraintsare concave functions; and (iii) the functions used to define any equalityconstraints are linear (or affine). If we have a concave program withonce-differentiable objective and constraint functions, then it turns out that


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Introductioncertain conditions due to Karush (1939) and Kuhn and Tucker (1951)involving the gradient vectors of the objective and constraint functionsevaluated at a point x* as well as certain (Lagrange) multipliers are suf f icientto imply that x* solves the concave programming problem; see Theorem2.30. In addition, if a relatively mild constraint qualification condition issatisfied, then these same Karush-Kuhn-Tucker conditions are alsonecessary for x* to solve the constrained maximization problem; seeTheorem 2.29. The multipliers that appear in the Karush-Kuhn-Tuckerconditions can often be given physical or economic interpretations: themultiplier (if unique) that corresponds to a particular constraint can beinterpreted as the incremental change in the optimized objective functiondue to an incremental relaxation in the constraint. For further details andrigorous statements of this result, see Samuelson (1947, p. 132), Armacostand Fiacco (1974), and Diewert (1984). Another result in Section 2.5,Theorem 2.28, shows that a concave programming problem has a solutionif a certain Lagrangian saddle point problem (which is a maximizationproblem in the primal variables and a minimization in the dualmultiplier variables) has a solution. This theorem, due originally to Uzawa(1958) and Karlin (1959), does not involve any differentiability conditions;some economic applications of it are pursued in the last section ofChapter 4.Chapter 3 deals with generalized concave functions; i.e., functions thathave some of the properties of concave functions but not all.

Section 3.1 defines the weakest class of generalized concave functions,namely, the class of quasiconcave functions. A function (defined over aconvex subset of Euclidean n- dimensional spacethroughout the book wemake this domain assumption) is quasiconcave if the values of the functionalong the line segment joining any two points in the domain of definitionof the function are equal to or greater than the minimum of the functionvalues at the end points of the line segment. Comparing the definition ofa quasiconcave function with the definition of a concave function, it canbe seen that a concave function is quasiconcave (but not vice versa). Recallthat concave functions played a central role in optimization theory becauseof their extremum properties. Quasiconcave functions also have a usefulextremum property, namely: every strict local maximizer of a quasiconcavefunction is a global maximizer (see Proposition 3.3). Quasiconcave functionsalso play an important role in the generalized concave mathematical program-ming problem (see Section 3.6), where the concave inequality constraintsthat occurred in the concave programming problem of Section 2.5 becomequasiconcave inequality constraints. Finally, quasiconcave functions playa central role in economic theory since the utility functions of consumersand the production functions of producers are usually assumed to bequasiconcave functions (see Chapter 4 below).


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Sections 3.1 and 3.2 provide various alternative characterizations ofquasiconcavity in a manner that is analogous to the alternative characteriz-ations of concavity that were developed in the opening sections of Chapter2. Three alternative characterizations of quasiconcavity are as follows: (i)the uppe r level sets of the function are convex sets for each level (D efinition3.1); (ii) if the directional derivative of the function in any fea sible directionis negative, then function values in that direction must be less than thevalue of the function evaluated at the initial point (this is the contrapositiveto Theorem 3.11); and (iii) the Hessian matrix of second-order partialderivatives of the function evaluated at eac h point in the domain of definitionis negative semidefinite in the subspace orthogonal to the gradient vectorof the function evaluated at the same point in the domain of definition(Corollary 3.20). Characterization (ii) above requires once differentiabilityof the function, while characterization (iii) requires twice continuousdifferentiability over an open convex set and the existence of a nonzerogradient vector at each point in the domain of definition. The restrictionthat the gradient vector be nonzero can be dropped (see Theorem 3.22),but the resulting theorem requires an additional concept that probably willnot be fam iliar, nam ely, the concept of a semistrict local minimum, explainedin Definition 3.3. This concept is also needed to provide a characterizationof semistrictly quasiconcave functions in the twice-differentiable case; seeTheorem 3.22. On the other hand, the familiar concept of a local minimumis used to provide a characterization of strictly quasiconcave functions inthe twice-differentiable case; see Theorem 3.26. In fact, all of the differenttypes of generalized concave functions can be characterized by their localminimum or maximum behavior along line segments; see Diewert, Avriel,and Zang (1981) for the details.Section 3 .4 deals with the p roperties and uses of the class of semistrictlyquasiconcave functions. A function is semistrictly quasiconcave iff for everytwo points in the domain of definition such that the function has unequalvalues at those two points, then the value of the function along the interiorof the line segment joining the two points is greater than the minimum ofthe two end-point function values; see Definition 3.11. If the function iscontinuous (or merely upper semicontinuous so that its upper level sets areclosed), then a semistrictly quasiconcave function is also quasiconcave(Proposition 3.30). It is easy to verify that a concave function is alsosemistrictly quasiconcave. Hence, in the continuous (or upper semicon-tinuous) case, the class of semistrictly quasiconcave functions lies betweenthe concave and quasiconcave classes. An alternative characterization ofthe concept of semistrict quasiconcavity for continuous functions in termsof level set properties is given b y P roposition 3.35: the family of upper levelsets must be convex and each nonmaximal level set must be contained in


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Introductionthe boundary of the corresponding upper level set (a maximal level setobviously must coincide with the corresponding upper level set). Semistrictlyquasiconcave functions have the same extremum property that concavefunctions had, namely: any local maximizer for a semistrictly quasiconcavefunction is a global maximizer (Theorem 3.37). Semistrictly quasiconcavefunctions also play a role in consumer theory; see Section 4.5.A more restrictive form of generalized concavity than semistrictquasiconcavity is strict quasiconcavity, discussed in Section 3.3 A functionis strictly quasiconcave if for every two distinct points in the domain ofdefinition of the function the value of the function along the interior of theline segment joining the two points is greater than the minimum of the twoend-point function values; see Definition 3.8. It is easy to verify that astrictly concave function is strictly quasiconcave and that a strictly quasicon-cave function is semistrictly quasiconcave and quasiconcave. Strictlyquasiconcave functions have the same extremely useful extremum propertythat strictly concave functions had: any local maximum is the unique globalmaximum. Strictly quasiconcave functions also play an important role ineconomics; see Section 4.6. Continuous strictly quasiconcave functions havestrictly convex upper level sets (Proposition 3.28).Section 3.5 deals with three new classes of generalized concave func-tions: (i) pseudoconcave, (ii) strictly pseudoconcave, and (iii) stronglypseudoconcave. These classes of functions are generalizations of the classof concave, strictly concave, and strongly concave functions, respectively.The three new classes of functions are usually defined only in the differenti-able case (although nondifferentiable definitions exist in the literature andare referred to in the text).A pseudoconcave function may be defined by the following property(the contrapositive to Definition 3.13): if the directional derivative of thefunction in any feasible direction is nonpositive, then function values inthat direction must be less than or equal to the value of the function evaluatedat the initial point. Pseudoconcave functions have the same importantextremum property that concave functions had: if the gradient vector of afunction is zero at a point, then that point is a global maximizer for thefunction (Theorem 3.39). A characterization of pseudoconcave functionsin the twice continuously differentiable case is provided by Theorem 3.43.

A strictly pseudoconcave function may be defined by the followingproperty (the contrapositive to Definition 3.13): if the directional derivativeof the function in any feasible direction is nonpositive, then the functionvalues in that direction must be less than the value of the function evaluatedat the initial point. Strictly pseudoconcave functions have the same impor-tant extremum property that strictly concave functions had: if the gradientvector of a function is zero at a point, then that point is the unique global


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6hapter 1maximizer for the function (Theorem 3.39). A characterization of strictlypseudoconcave functions in the twice continuously differentiable case isprovided by Theorem 3.43.S trongly pseudoconcave funct ions are strictly pseudoconcave functionswith the following additional property: if the directional derivative of thefunction in any feasible direction is zero, then the function diminishes(locally at least) at a quadratic rate in that direction. Recall that in the twicedifferentiable case, a strongly concave function could be characterized byhaving a nega tive definite Hessian m atrix o f second-order partial derivativesat each point in its domain of definition. In the twice differentiable case, afunction is strongly pseu doconcave if its He ssian matrix is nega tive definitein the subspace orthogonal to the gradient vector at each point in the domainof definition (Proposition 3.45). The property of strong pseudoconcavity issometimes called strong quasiconcavity in the economics literature, andsome economic applications of this concept are developed in Section 4.7.It should b e noted that all of our concavity and q uasiconcavity concep tshave convex and quasiconvex counterparts: a function f is convex (q uasicon-vex) iff f is concave (quasiconcave).

C hapter 3 is concluded by S ection 3.6, which deals with generalizationsof the concave programs studied in Section 2.5. An example shows that theKarlin-Uzawa Saddle Point Theorem for (not necessarily differentiable)concave programming problems cannot be readily generalized. However,for differentiable programs, the sufficiency of the Karush-Kuhn-Tuckerconditions for concave problems can be generalized to programming prob-lems involving objective and constraint functions that satisfy some type ofgeneralized concavity property: Theorem 3.48 shows that the objectivefunction need only be pseudoconcave, the equal to or greater than inequalityconstraint functions need only be quasiconcave, and the equality constraintfunctions need only be quasimonotonic. A function is quasimonotonic if itis both quasiconcave and quasiconvex (inequality 3.35). Thus thesepseudoconcavity, quasiconcavity, and quasimonotonic properties replacethe earlier concavity and linearity properties that occurred in Theorem 2.30.Chapter 4 deals with economic applications. We consider four modelsof economic behavior: (i) a producer's cost minimization problem, (ii) aconsumer's utility maximization problem, (iii) a producer's profit maximiz-ation problem, and (iv) a model of national product maximization for aneconomy that faces world prices for the outputs it produces and is con-strained by domestic resource availabilities. In the context of the abovefour models, we show how each of the types of generalized concavity studiedin Chapters 2 and 3 arises in a natural way.C hapter 4 also proves some econom ics duality theorem s. M any problemsin economics involve maxim izing or minimizing a function subject to another


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Introductionfunctional constraint. If either the objective function or the constraintfunction is linear (or affine), then the optimized objective function may beregarded as a function of the parameters or coefficients (these are usuallyprices) of the linear function involved in the primal optimization problem.This optimized objective function, regarded as a function of the pricesappearing in the primal problem, is called the dual function. Under certainconditions, this dual function may be used to reconstruct the nonlinearfunction that appeared in the primal optimization problem. The regularityconditions always involve some kind of generalized concavity restrictionson the nonlinear primal function. Some applications of these economicsduality theorems are provided in Chapter 4.Chapters 5 and 6 deal with the following important question: how canwe recognize w hether a given function has a generalized concavity property?In the first part of Chapter 5, composition rules for the various typesof generalized concave functions are derived. Suppose we know that certainfunctions have a generalized concavity property (or are even concave). T henunder what conditions will an increasing or decreasing function of theoriginal function or functions have a generalized concavity property?In the second part of Chapter 5, we apply these composition rules toderive conditions under which a product or ratio of two or more functionshas a generalized concavity property, provided that the original functionsare concave or convex. Special attention is given to the case of productsand ratios of only two functions. The material in this chapter draws heavilyon the work of Schaible (1971, 1972).Chapter 6 deals with the generalized concavity properties of an impor-tant class of functions, namely, the class of quadratic functions. It turnsout that restricting ourselves to the class of quadratic functions simplifieslife somewhat: quasiconcave and semistrictly quasiconcave quadratic func-

tions cannot be distinguished. Furthermore, strictly and strongly pseudocon-cave quadratic functions cannot be distinguished. However, even with thesesimplifications, the characterization of the generalized concavity propertiesof quadratic functions proves to be a rather complex task. C hapter 6 developsall known results using a unified framework (based on the compositefunction criteria developed in Chapter 5) on the generalized concavityproperties of quadratic functions. Furthermore, many of the criteria areexpressed in alternative ways using eigenvalues and eigenvectors or deter-minantal conditions. The material in this chapter summarizes and extendsthe work of Schaible (1981a,b).Chapter 7 provides a brief survey of fractional programming and indi-cates how generalized concavity concepts play a role in this importantapplied area. A fractional program is a constrained maximization problemwhere the objective function is a ratio of two functions, say f(x)/g(x), and


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Chapter 1

the decision variables x are restricted to belong to a closed convex set S infinite-dimensional Euclidean space. In a linear fractional program , the func-tions f and g are both restricted to be linear or affine. In a concav e fractionalprogram, the numerator function f is restricted to be nonnegative andconcave and the denom inator function is restricted to be convex and positiveover the constraint set S. In a gene raliz ed f ract ional program , we maximizea sum of ratios or we maximize the minimum of a finite number of ratios.In Section 7.1, we show that the objective function in a concavefractional programming problem is semistrictly concave. Hence, a localmax imum for the prob lem is a global maximum . If, in addition, the objectivefunction in a concave fractional program is differentiable, we show that theobjective function is pseudoconcave. In this latter case, the Karush-Kuhn-Tucker conditions are sufficient (and necessary if a constraint qualificationcondition is satisfied) to characterize the solution to the fractional program -ming problem.

Section 7.2 surveys a numb er of applications of fractional program ming .Business and economics applications of fractional programminginclude the following:1 . Maximization of productivi ty . The productivity of a firm, enterprise,

or economy is usually defined as a function of outputs produceddivided by a function of the inputs utilized by the firm.2 . Max imization of the rate of return on investments.3. M inim izat ion of cost per uni t of t ime.4 . M ax imization of an econom y 's grow th ra te . This problem originatesin von Neumann's (1945) model of an expanding economy. Theoverall grow th rate in the econ omy is the smallest of certain sectoralgrowth rates. Maximizing the minimum of the sectoral growth ratesleads to a generalized fractional programming problem.5. Portfolio selection problem s in f inance. Here w e attemp t to maximizethe expected return of a portfolio of investments divided by the riskof the portfolio.Applied mathematics applications of fractional programming includethe follow ing:1. Finding the maximal eigenvalue. The maximal eigenvalue A of asymmetric matrix A can be found by maximizing the ratio of two

quadratic forms, i.e., A = max., {x TAx/x Tx: x 0 0}.2. A pprox im ation theory . Some problems in numerical approximationtheory generate generalized fractional programs.3. Solution of large-scale linear programs. Using decompositionmethods, the solution to a large linear program can be reduced to


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Introductionthe solution of a finite number of subproblems. These subproblemsare linear fractional programs.4. Solving stochastic programs. Certain stochastic linear programmingproblems lead to fractional programming problems. This class ofapplications includes the portfolio selection problem mentionedabove.The above applications of fractional programming (and additionalones) are discussed in Section 7.2 and references to the literature areprovided there.In Section 7.3, we indicate how a concave fractional program may be

transformed into a family of ordinary concave programs using a separationof variable technique. However, an even more convenient transformationis available. Propositions 7.2 and 7.3 show how concave fractional program scan be transformed into ordinary concave programs using a certain changeof variables transformation. We also derive the (saddle point) dual program-ming problems for a concave fractional program in this section.Section 7.4 concludes C hapter 7 b y outlining some possible algorithmicapproaches to the solution of concave fractional programs.The material in Chapter 7 draws heavily on Schaible (1978, 1981c).C hapter 8 introduces two new classes of generalized concave functions:transconcave functions and (h, 4)- concave functions.A functionf efined over a convex subset C of Euclidean n-dimensional space is t ransconcave (or G - concave) if it can be transformedinto a concave function by means of a monotonically increasing functionof one variable G; hence f is G-concave if h(x) = G[f(x)] is a concavefunction over C.Transconcave functions are used in at least two important areas ofapp lication. The first use is in num erical algorithms for m aximizing functionsof n variables; if the objective function f n the nonlinear programmingproblem can be transformed into a concave function G[f(x)] by means ofan increasing function of one variable G , then the original ob jective functionf(x) may be replaced by the concave objective function G[f(x)] and oneof many concave p rogramming algorithms m ay be used to solve the problem.A secon d use is in the comp utation of general equilibria in econom ic modelswhere the number of consumers in the model is smaller than the numberof commodities. In order to compute a general equilibrium (see Debreu,1959, for a formal definition and references to the literature), an algorithmis required that will compute a fixed point under the hypotheses of theKakutani (1941) Fixed Point theorem. Scarf (1967) has constructed suchan algorithm, but it is not efficient if the number of commodities exceeds50. However, if the preferences of all consumers in the general equilibrium


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10hapter Imodel can b e represented by m eans of concave utility functions, then Neg ishi(1960) showed how a general equilibrium could be computed by solving asequence of concave programming problems, where the objective functionin each prob lem w as a w eighted sum of individual utility functions. D iewert(1973) later showed how the Negishi framework could be simplified to aproblem where it is necessary to find a fixed point in the space of utilityweights. Hence, for an economic model where the number of consumerclasses is smaller than the number of commodity classes, it will be moreefficient to compute the fixed point over the space of nonnegative utilityweights rather than over the space of nonnegative commodity prices.It is obvious that transconcave functions mu st be at least quasiconcave.In fact, Proposition 8.1 shows that transconcave functions must be semi-strictly quasiconcave. An example shows that the property of semistrictquasiconcavity is not sufficient to imply transconcavity. In the case ofdifferentiable functions, Proposition 8.7 shows that transconcave functionsmust be pseudoconcave. Furthermore, Theorem 8.25 shows that a twicecontinuously differentiable strongly pseudoconcave function is transcon-cave. H ence, in the tw ice differentiable case, necessary and sufficient condi-tions for transconcavity lie between the properties of pseudoconcavity andstrong pseudoconcavity.In Section 8.2, we derive necessary and sufficient conditions for trans-concavity in the twice differentiable case. Even in this differentiable case,the conditions are somewhat complex. [We do not attempt to treat thenondifferentiable case, which has been treated by Kannai (1977, 1981) insome detail.] The first necessary condition for transconcavity is pseudocon-cavity. The next necessary condition is the existence of a negative semi-definite augmented Hessian matrix (Condition B). Let us explain what thiscondition means when n = 1, i.e., when f(x) is a function of one variable.In this case, G[f(x)] is a concave function over the convex set C if thesecond derivative G'[f(x)]f"(x) + G"[f(x)][ f (x)]2 0 for every x belong-ing to C. Since we restrict ourselves to functions G such that G'[f(x)] > 0,we require f"(x) r(x)[f'(x)]2 - 0, where r(x) = G"[f(x)]/G'[f(x)]. Inthis n = 1 case, Condition B becomes f'(x) = 0 implies f"(x) : 0; i.e., ifthe first derivative o f f is zero at a point x belonging to C, then the secondderivative is nonpositive. This condition is already implied by ConditionA, p seudoconcavity of f, when n = 1. For a general n, C ondition B becomes:there exists a scalar r(x) such that V 2f(x) r(x)Vf(x)Vf(x) T is negativesemidefinite for each x belonging to C, where V 2f(x) is the n x n matrixof second-order partial derivatives of f evaluated at x = (x...... x) T andVf(x) is the n-dimensional column vector of first-order partial derivativesof f evaluated at x. This condition may be phrased as a problem in matrix


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IntroductionIalgebra: under what conditions on A = V 2f(x) and b = Vf(x) does thereexist a scalar r such that A rbb T is negative semidefinite? We answer thisquestion in great detail in Section 8.2, providing several alternative andequivalent answers using bordered Hessian matrices, restricted eigenvaluecriteria, and various determinantal conditions. This material may be ofindependent interest from the viewpoint of matrix algebra. (It should bementioned that these matrix criteria assume that n > 2.) Given that an rexists that makes A rbb T negative semidefinite, we derive three equivalentmethods for computing the minimal r that will do the job; some of thismaterial has not been published before. The two final necessary conditionsfor transconcavity of f in the twice-differentiable case, Conditions C andD, involve the boundedness of the minimal r(x) such that V 2f(x) r(x)Vf(x)Vf(x) T is negative semidefinite over x belong ing to C . In the casen = 1, we require that sup, if"(x)/[ f'(x)]`: f'(x) # 0, x E C} be finite. Thislast condition is not implied by Condition A or B. Theorem 8.18 shows thatConditions A-D are necessary and sufficient for transconcavity.Definition 8.2 defines a special class of transconcave or G-concavefunctions, namely, the class of r-concave functions. A function f definedover a convex set C is r-concave ifff s G-concave with G(t) = e". Thisspecial class of transconcave functions plays an imp ortant role in S ection 8.2.The first two sections of Chapter 8 deal with the class of transconcaveor G-concave functions. These are functions that can be transformed intoconcave functions by means of a monotonic transformation G of the imagesof the original function f The last section of Chapter 8 deals with a morecomplicated class of functions, the class of (h, 1)-concave functions. Afunctionf s (h, d)-concave iff for every x' and x 2 belonging to C andscalar A between 0 and 1, we have 4)[f{h - '[(1 A)h(x') + Ah(x 2 )]}] _(1 A) 44 f(x' )] + A4[ f(x 2 )], where j is an increasing, continuous functionof one variable and h(x) = {h,(x), ... , h(x)} is a vector-valued functionof n variables x = (x, , ... , x,,) that is continuous and one-to-one over C.We also assume that the image set h(C) is convex. Theorem 8.30 showsthat f is (h, 0)-concave over the set C iff f(v) = 0[ f{h - '(v)}] is a concavefunction over the set h(C), which we assume convex. Thus, f s (h, 4)concave if we can find a one-to-one and continuous function h such thatf{h-(y)} is a quasiconcave function and an increasing continuous functionof one variable that further transforms f{h '} into the concave function^[ f{h '}]. It can be verified that a function f defined over a convex set Cis (i) concave iff it is (h, )-concave with h(x) = x and 4(t) = t, (ii)G-concave iff it is (h, (h)-concave with h(x) = x and 0(t) = G(t), and (iii)r-concave if it is (h, 0)-concave with h(x) = x and q(t) = exp (rt).Thus, the class of (h, q5)-concave functions contains our earlier classes of


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12hapter 1transconcave functions. Another interesting class of (h, 4)-concave func-tions is the class of (log, log)-concave functions. In this case, h(x )[h,(x,, ... , x), ... , h(x,, ... , x)] = [ log x...... log x] and 0(t) = log t.Functions that are (log, log)-con cave ap pear in a special branch of nonlinearprogramming called geometric programming.Theorem 8.31 in Section 8.3 shows that differentiable (h , )-concavefunctions f have the same property that concave (and pseudoconcave)functions had; i.e., if there exists an x* such that Vf(x*) = 0, then x* isthe global maximizer f or f. For a characterization of (h, 0)- concave func-tions in the twice-differentiable case , as well as some app lications of (h, 4)-concavity to statistical decision making, see Ben-Tal (1977).

C hapter 9 p rovides an introduction to another two classes of gen eralizedconcave functions: the class of F-concave functions due to Ben-Tal andBen-Israel (1976 ), and the class of arcw ise connected functions due to Ortegaand Rheinboldt (1970) and Avriel and Zang (1980).In Chapter 2, it is shown that a differentiable concave function definedover an open convex set has the property that the first-order Taylor seriesapproximation to the function around any point lies above the graph ofthe function, and moreover, this supporting hyperplane property can serve

to characterize differentiable concave functions. Theorem 9.1 shows thatthis supporting hyperplane property can serve to characterize concavefunctions even in the nondifferentiable case. A natural generalization ofconcavity is obtained by considering functions whose graphs are supportedfrom above by functions that are not necessarily linear. These supportfunctions are drawn from a prespecified class of functions denoted by theset F. If F is defined to be the class of affine functions, then the class ofF- concave functions reduces to the class of concave functions. In Section9.1, we present several interesting examples of F- concave functions forvarious definitions of the supporting class of functions F.Section 9.2 discusses various families of arcwise connected functions.In the definitions of the various kinds of concave and generalized concavefunctions, the behavior of the function along the straight line segmentjoining any two points in the domain of definition was restricted in someway. W e define quasiconnec ted, sem istrictly quasiconne cted, strictly quasicon-nected, pseudoconnected, and strict ly pseudoc onnected functions in a manneranalogous to the definitions of quasiconcavity, semistrict quasiconcavity,strict quasiconcavity, pseudoconcavity, and strict pseudoconcavity, respec-tively, by replacing the straight line segments that occur in the latterdefinitions by continuo us arcs. M oreover, the old convex domain of definitionset is replaced by an arcw ise connected set , and convex upper level sets arereplaced by arcwise connected upper level sets. The extremal properties ofthe new classes of "connected functions turn out to be virtually identical


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Introduction3to the extremal properties of the corresponding class of quasiconcavefunctions. Since the "connected" classes of functions are much more g eneral,the reader may well wonder why we devote so much space to the variousclasses of generalized concave functions and so little space to the "con-nected" functions. An an swer is that it is much easier to characterize concaveand generalized concave functions. For example, in the twice differentiablecase, concave functions m ay b e characterized by the local property of havinga negative semidefinite Hessian matrix at each point of the domain ofdefinition. A comparable local characterization for the various classes of"connected" functions does not appear to exist. Furthermore, there are nocomposition rules for "connected" functions that are analogous to thecomposition rules developed for generalized concave functions in Chapter5. However, on the positive side, the reader will see that it is not very muchmore difficult to develop the theory of "connected" functions than it is todevelop the properties of the corresponding classes of quasiconcave func-tions.We conclude by noting that more detailed references to the literatureon generalized concavity will appear at the ends of the respective chaptersand in the Supplementary Bibliography.

ReferencesARMACOST, R.L., and FlACCO, A.V. (1974), Computational experience in sensitivity analysisfor nonlinear programming, Math. Programming 6, 301-326.AVRIEL, M., and ZANG, 1. (1980), Generalized arcwise connected functions and characteriz-ations of local-global minimum properties, J. Opt. Theory Appl. 32, 407-425.BEN-TAL, A. (1977), On generalized means and generalized convexity, J. Opt. Theory Appl.21, 1-13.BEN-TAL, A., and BEN-ISRAEL, A. (1976), A generalization of convex functions via support

properties, J. Australian Math. Soc. 21, 341-361.DEBREU, G. (1959), Theory of Value: An Axiomatic Analysis of Economic Equilibrium, JohnWiley, New York.DIEWERT, W.E. (1973), On a theorem of Negishi, Metroeconomica 25, 119-135.DIEWERT, W.E. (1984), Sensitivity analysis in economics, Comput. Oper. Res. 11, 141-156.DIEWERT, W. E., AVRIEL, M., and ZANG, 1. (1981), Nine kinds of quasiconcavity andconcavity, J. Economic Theory 25, 397-420.KAKUTANI, S. (1941), A generalization of Brower's fixed point theorem, Duke Mathematical

Journal 8,457-459.KANNAI, Y. (1977), Concavifiability and the construction of concave utility functions, J. Math.

Econ. 4, 1-56.KANNAI, Y. (1981), Concave utility functionsExistence, constructions and cardinality, in

Generalized Concavity in Optimization and Economics, Edited by S. Schaible and W.T.Ziemba, Academic Press, New York, pp. 543-611.


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1 4h a p t e r 1KARLIN, S. (1959), M athem atical Methods and Theory in Games, Program m ing and Econom ics,Vol..1, Addison-Wesley, Reading, Massachusetts.KARUSH, W. (1939), Minima of functions of several variables with inequalities as sideconditions, M.Sc. dissertation, Department of Mathematics, University of Chicago.KUHN, H.W., and TUCKER, A.W. (1951), Nonlinear programming, in Proceedings of th eSecond Berkeley Symposium on Mathematical Statistics and Probability, Edited by J.Ney matl, University of California Press, Berkeley, pp. 481 -492.

NEGISHI, T. (1960), Welfare economics and existence of an equilibrium for a competitiveeconomy, M etroeconom ica 12, 92-97.

ORTEGA, J.M., and RHEINBOLDT, W.C. (1970), Interactive Solution of N onlinear Equat ionsin Sev eral V ariables, Academic Press, New York.SAMUELSON, P.A. (1947), Foundation of Economic Analysis, Harvard University Press,C ambridge, M assachusetts.SCARF, H. (1967), The approximation of fixed points of a continuous mapping, S I A M J. A p p l.Math. 15, 1328-1343.SCHAIBLE, S. (1971), Beitriige zur quasikonvexen Programmierung, Doctoral Dissertation,

Kln.SCHAIBLE, S. (1972), Quasiconvex optimization in general real linear spaces, Z. Oper. Res.

16, 205-213.SCHAIBLE, S. (1978), A nalyse and A nw endungen von Quot ien ten Program m en, Hain-Verlag,

Meisenheim.SCHAIBLE, S. (1981a), Quasiconvex, pseudoconvex and strictly pseudoconvex quadratic

functions, J. Opt . T heory A ppl . 35, 303-338.SCHAIBLE, S. (1 981b ), Generalized convexity of quadratic functions, in Generalized Concavityin Opt im ization and Econo m ics, Edited by S. Schaible and W.T. Ziemba, Academic Press,

New York, pp. 183-197.SCHAIBLE, S. (1981c), A survey of fractional programming, in Generalized Con cavity inOpt imiza t ion and Econom ics, Edited by S. Schaible and W.T. Ziemba, Academic Press,New York, pp. 417-440.UZAWA, H. (1958), The Kuhn-Tucker theorem in concave programming, in Studies in L inearand Nonlinear Programming, Edited by K. Arrow, L. Hurwicz, and H. Uzawa, StanfordUniversity Press, Stanford, California, pp. 32-37.VON NEUMANN, J. (1945), A model of general economic equilibrium, R ev. Econ . Stud. 13, 1-9.


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ConcavityIn this chapter we present some basic results on concave functions that willbe generalized in subsequent chapters. Since the discussion of generalizedconcavity in this book is oriented toward applications in economics andoptimization, only selected properties of concave functions pertinent to theabove applications are mentioned here. The origins of the concepts ofconvex sets and concave and convex functions can be traced back to theturn of the century; see Holder (1889), Jensen (1906), and M inkowski (191 0,1911).

2.1. Basic Definitions

The domain of a concave function is always a convex set. We thereforebegin our formal discussion of concavity with the following definition.Definition 2.1. A subset C of the n-dimensional real Euclidean spaceR" is a convex set if for every x` EC, x 2 E C and 0


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16hapter 2values; we nevertheless sacrifice generality for the sake of simplicity. Wethen have the following definition.Definition 2.2. A function f defined on the convex set C cR" is calledconcave if for every x' E C , x 2 E C, and 0


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Concavity7such that B is between A and C, then B is on or above the line segment(chord) AC. Letting A = f(x'), B = f(x'), and C = f(x z ), the reader caneasily verify the following relationship for a concave function that holdsfor x' 0, where k 1 , k, are given positive numbers. This function may

represent the cost of manufacturing some commodity, where k, is the fixedcharge or setup cost, which is incurred only when the commodity is actuallybe ing manufactured. Th is function is strictly concav e, having a discontinuityat a boundary point of its domain.

In general, concave functions are continuous in the interior of theirdomain but may have discontinuities at boundary points. In particular,concave functions are continuous on an open interval, a result we shownext. A function f is said to be Lipschitz continuous (Bartle, 1976) on aninterval [a, b] if for any two points x c [a, b], y c [a, b] there exists aconstant K such that

f(x) f(y) j ^ KIx y^.2.3)f ( x )( x )( x )/(a)b )c )x1rFigure 2.2. Concave functions.


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1 8hapter 2It is easy to see that a Lipschitz continuous function is continuous.Proposition 2.1 (Roberts and Varberg, 1973). Let f be a concave func-tion on the convex set C c R. Then f is Lipschitz continuous on everyclosed interval contained in the interior of C, and consequently, f iscontinuous on the interior of C.Proof. First, let [a, b]c C be a closed interval and let m =min { f(a), f(b)}. Then, for any x = A a + (1 A )b, 0 _ A f ( a) + ( 1 A ) f (b ) ? A m + ( 1 A ) m = m .2.4)Thus f is bounded from below by m on [a, b]. Similarly, let (a b)/2


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Concavity9f is nonconstant on [a s, b + E], hence m < M l. Let x E [a, b], y E [a, b],x # y, and definez=y (xy),_ Ixy)(29Iy xI+fixy1,Then, z E [a r, b + e], y = Az + (1 A)x, andf(y) ^ Af(z) + (1 A)f(x) = A[f(z) f(x)] +f(x).2.10)Hence,

f(x) f(y) 75 A[f(x) f(z)]A(M n) < k y' (M m )2.11)Eand, since x and y are arbitrary points in [a, b], we conclude thatf(x) f(y)l


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2 0hapter 2Example2.3. Letf(x)=xfor0


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Concavity1It is strictly concave if and only if the ineq uality in (2.17) is strict for x 0 x .Proof. Suppose that f is concave. Without loss of generality assumethat x < x . Let t be a sufficiently small positive number such that x + t E C.Then, by the definition of concave functions and by letting x _

Ax + (1 A)(x + t) we get0

f(x) x +txf(x)+ x +t xxf(x+t).2.18)Subtracting f(x) from both sides of (2.18) and dividing by x x yields

f(x ) -f(x) f(x + t) -f(x) f(x+ t) -f(x)(219)xx+txLetting t - 0 we obtain xf( X) f(x) ^ f'(x2.20)-yielding (2.17) by a few simple algebraic manipulations.Conversely, let x' and x 2 be any two points in C and let x' _Ax' +(1 A )x z for some 0< A < 1. Then, by (2.17)

f(x') -5 f(x') + (I A)f'(x 3 )(x' x2 )2.21)f(x2 ) .T(X3 ) + Af (x3 )(x` x').2.22)Multiplying (2.21) and (2.22) by A and (1 A), respectively, and addingup, we get

f(x3 ) = f(Ax' + (1 A)x 2 ) > Af(x') + (1 A)f(x 2 )2.23)for 0


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2 2hapter 2f ( X )

X Figure 2.4. The "graph below the tangent"property of concave functions.

Theorem 2.3 (Fenchel, 1951). Let f be a differentiable function on theopen convex set C c R. It is concave (strictly concave) if and only if f' isa nonincreasing (decreasing) function.Proof. Suppose f is (strictly) concave and let x', x 2 be two points in

C such that x' < x 2 . Then, by (2.17)f(x2 ) f(x') , f(X 2 )2.24)xz -x 'a n d f(x2) f(x') .2.25)

x xHence

f(x') f(x2 )2.26)and f' is nonincreasing. The inequalities in (2.24)-(2.26) are strict if f sstrictly concave. Conversely, let x' E C, x 2 c C, x' < x 2 and x' _Ax' + (1 A )x 2 for some 0 < A < 1.

By the Mean Value Theorem (see Bartle, 1976),f(x) = f(X3 ) + Af'(x)(x` x'),3 < X < X2 227f(x3 ) = f(x`) + (1 - A)f'(z)(x 2 - x'),' < z < x3 .2.28)


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Concavity3If f' is nonincreasing, we havef(x3 ) ?f(x') + (I ,1)f (x)(x 2 x')2.29)

and if f' is decreasing, the inequality in (2.29) is strict. Multiplying (2.27)and (2.29) by (A 1) and A , respectively, and adding up, we obtainAf(x 3 ) + (,1 1)f(x 2 ) ? Af(x I ) + (,1 1)f(x 3 )2.30)or

(x3) =f(dx' +(I 1)x) > Af(x') + (1 A)f(x 2 ).2.31)Thus, if f' is nonincreasing (decreasing), then f is concave (strictly conc ave).

Note that instead of saying that f' is nonincreasing, we could also saythat for every two points x EC, x z c C[f'(x 2 ) f(x')](x 2 x')


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24hapter 22 .3. Concave F unctions of Several VariablesMost of the results of this section are straightforward extensions of theone-dimensional case. Before w e present them, however, let us view concavefunctions from a new angle.We have seen that concave functions and convex sets are related: thedomain of a concave function is a convex set. Another relationship b etweenconcave functions and convex sets is given below. Let f be a functiondefined on the convex set C c R. The set

H(fj=((x,a:xE CaE 9,f(x)?a (2.33)inRs called the hypograph off t is the set of all points lying on orbelow the graph of f Similarly, the epigraph of f is the set

P(f)={(x,a):xECaE Rf(x)


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Concavity5Thus the point(x, a) = (Ax' + (1 A)x 2 , Aa' + (1 A)a 2 )2.36)

is also in H(f). Conversely, if H(f) is a convex set, then for every twopoints (x', a') E H(f), (x2 , a 2 ) E H(f) and 0 _ Af(x') + (I A)f(x 2 ) ,2.37)andf s concave.For any function fon C and any a E R the set U(f, a) defined by

U(f,a)={x:xc C,f(x)_a}2.38)is called the upper level set off Similarly, the setL(f,a)={x:xEC,f(x)


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2 6hapter 2f ( X )X

Figure 2.6. Upper-level set.

Proposition 2.7. The function f, defined on the convex set C c R", isconcave if and only if GU(f, 6, a) is a convex set for every 6 E R" and a c R.Proof. Suppose that f s concave and let x' c GU( f, 6 , a ), x z cGU(f, f , a ) for some ^ a . Then

f(x') ? (^) Tx' + a 241f ( x )

a\^/GU (f,e ,a e )

X

Figure 2.7. A generalized upper-l e v e l s e t .


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Concavity7f(x) > (C) x`' + a (242a n d

f(Ax' + (1 A)x z )? Af(x') + (1 A)f(x z ) .2.43)It follows from (2.41)-(2.43) thatf(Ax'+(1A)xz)?A[(6) Tx! +a]+(1A)[(6 ) Tx z +a]2.44)_ ( 0 ) T [Ax' + (1 A)x z ] + a .2.45)Hence GU( f, ^ a ) is a convex set.Conversely, let GU(f, , a) be convex for every 6 E R" and a c R. Letx' and x z be two distinct points of C, the convex domain off. Without lossof generality assume that x 0 x;. Suppose that f is not concave, that is,for some 0< A < 1, and X = ,1x' + (1 A)x 2 we have(z)


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28hapter 2where (2.53) follows from (2.46). Hence z 0 GU(f, 4 , a ). But(6 0 ) Tx' + a = f(x')2.54)(6 ) Tx2 + a = f(x 2 )2.55)and x' E GU( f, f, a ), x 2 E GU( f, 4, a ), contradicting that GU(f, f , a )saconvexseWe have already seen in Example 2.2 that a concave function mayhave discontinuities at boundary points of its domain. It is easy to see thatthe hypograph of such a function is not a closed convex set.

Definition 2.4. A function f defined on the convex set C c R" is calleda closed concave function if its hypograph H(f) is a closed convex subsetof R" + '.It follows that if f is a closed concave function then its upper-levelsets U(f, a) are closed convex subsets of R" for every real a.Next we derive and prove a result stating that concave functions are

continuous on open sets. The line of proof follows that of Roberts andVarberg (1973). First we need the following lemma.Lem ma 2.8. Let f be a concave function on the open convex set C c R".If f is bounded from below in a neighborhood of one point x E C, then itis locally boundedthat is, each x E C has a neighborhood on which f s

b o u n d e d .Proof. We first show that if f is bounded from below in a neighborhoodof x , it is also bounded from above in the same neighborhood. Supposethat f is bounded from below by a number m in NF (x ) ={z: z E C , liz x ii < e}. We can express every z E N F (x ) as z = x + Oy,where y E R' is a vector such that IiyVV = 1 and 0 is a sufficiently smallpositive number. Then

x = z(x + B Y ) + i(x Oy)2.56)a n d

f(x) if(x + Oy) + ?f(X O Y ),2.57)2f(x) f(x Oy) > f(x + Oy).2.58)


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Concavity9By the hypothesis f(x B y ) > m , hence2f(x) m > f(x + 6y) = f(z)2 . 5 9 )

and f is bounded from above for every z E N F (x ) .Let x E C, x ^ x . Then x = x + ay, w here again y E R ", II Y II = I anda is a positive number. Choose p> a such that u = x + py E C and letA = a/p. Then,

NN (x)={v: V E C,v=(1,1)z+ Au, zE N F (x ) }2.60)is a neighborhood of x with radius S = (1 A)e. Also, for v E N N (x)

f(v) > (1 A)f(z) + Af(u) _> (1 , 1 ) m + , 1 f (u ) .2 . 6 1 )That is, f is bounded from below on NS (x), and by the first part of theproof,f s also bounded from above on NN (x).A functionf efined on an open set U c R" is said to be locallyLipschi tz continuous if at each x E U there is a neighborhood NF (x ) anda constant K(x ) such that if x' E U, x z E U, thenf(x') f(x 2 )I


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30hapter 2implying that f(x2 ) f(x 3 ) > 2M and hence f(x 3 )l > M, a contradiction,and f must be locally Lipschitz continuous at every x EC, hence continuousonCThe last result can be generalized to show that concave functions arecontinuous on the relative interior of their domain. To state this moregeneral result we need a few preliminaries.Let A be a subset of R" such that for every two points x' EA, x 2 EAand for every real number a, also ax' + (1 a )x 2 EA. Such a set is calledan aff ine set. Clearly, affine sets are convex. Examples of affine sets in R "are single points, lines, hyperplanes, and the whole space R. For a convexset C c R" the intersection of all affine sets containing C is called the affinehull of C. The relative interior of C, denoted by ri C, is the interior of Crelative to its affine hull. The interior of C (relative to R ") may be emptyalthough ri C is nonempty as in Example 2.4 below. Generally this is thecase if the affine hull is a proper subset of R. If the affine hull of C is R " ,then the interior of C is equal to the relative interior of C.

Example 2.4. Let C be the unit (two-dimensional) circle in three-dimensional space R'that is, letC={x: x cR, (x,)+ ( x 2 ) 2 < _ 1, x 3 = 0 } .2.65)This set has no interior (relative to R') since one cannot find an open sphe rein R' contained in C. The affine hull of C is the (x,, x 2 ) plane and

ri C = {x: x ER', (x,) 2 + ( x 2 ) 2 < 1, x 3 = 0} ,2.66)

that is, the interior of C relative to its affine hull.We can state now the following proposition.

Proposition 2.10. If f is a concave function defined on the convex setC c R" then it is continuous on ri C. In particular, if C is an open convexset then f is continuous on C.Proof of this theorem requires additional background material and willbe omitted here. Readers interested in the proof are referred to Fenchel

(1951) or Rockafellar (1970).An important property of concave functions useful in proving severalresults is that a function of n variables is concave if and only if it isconcave on every line segment included in its domain. Formally we havethe following.


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Concavity1Proposition 2.11. The function f defined on the convex set C R" isconcave if and only if for every x' c C, x 2 E C the single-variable functionF, defined byF(A) = f(Ax' + (1 A)x 2 )2.67)

is concave for every 0 A < 1.The proof is straightforward and is left for the reader. Proposition 2.11enab les us to extend several results holding for con cave func tions of a singlevariable to the n-variable case.Let us turn now to differentiable concave functions. First we need to

define differentiable functions on subsets of R.Definition 2.5. Let f e a function defined on the open set U c R"and let x E U. Thenf s said to be differentiable at x if there exists avector T(x ) such that for all x E R" satisfying x + x E U we have

f(x+ x) = f(x) + xTT(x) + a(x, x)IIxII2.68)where a is a real function such that lim x . a(x , x) = 0. Moreover, f is saidto be differentiable on U if it is differentiable at every x E U .As in the single-variable case, if f s differentiable at x then f scontinuous there and T(x ) is equal to Vf(x ), the gradient vector of f atx (see, e.g., Bartle, 1976), where

a f ( x ) / a x 1V f ( x ) _ 2 . 6 9 )af(x)/ax

The next result is a direct extension of Theorem 2.2.Theorem 2.12 (Avriel, 1976; Mangasarian, 1969). Let f be a differenti-able function on the open convex set C c R. It is concave if and only iffor every x c C, x E C we have

f(x) f(x ) + (x x ) T Vf(x ) .2.70)It is strictly concave if and only if the inequality in (2.70) is strict for x ^ x .

Proof. Let f be concave on C and let x E C, x E C, such that x ^ x.It follow s that for 0 < A < _ 1f(x + A (x x )) = f(Ax + (1 A )x ) > Af(x) + (1 A)f(x )

(2 .71)


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32hapter 2o r(x-f(x 1 /A)[f(x + A(x - x )) - f(x )] .2.72)Substituting (2.68) into the last inequality we obtain

f(x) - f(x ) f(x') (2.74)a n d

f(Ax' +(1- A)x) + A(x 2 - x')T Vf(Ax' + (1 - A )x 2) ? f(x 2 ).2.75)Multiplying (2.74) and (2.75) by A and (1 - A), respectively, and addingup we getf(Ax' + (1 - A)x 2 ) >_ Af(x') + (1 - A)f(x 2 )2.76)as asserted.To prove the result for the strictly concave case, suppose that f isstrictly concave and x E C, x E C, such that x 0 x . Then, sincef is concave,by (2.71)-(2.73) we have that (2.70) holds.We now show that equality cannot hold in (2.70). Suppose, to thecontrary, that

f(x) = f(x) + (x - x) T Vf(x).2.77)Then, for 0 < A < 1,f(Ax + (1 - A)x)> Af(x) + (1 - A)f(x)= f(x) + (1 - A)(x - x) T Vf(x),2.78)where the inequality follows from the strict concavity off and the equalityfrom (2.77). Now let x 2 = Ax + (1 - A)x. Then, since x2 E C andf sconcave w e obtain (2.70) w ith x 2 replacing x that is,

f(Ax + (1 - A)x) < f(x) + (1 - A)(x - x ) T Vf(x)2.79)


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Concavity3holds. However, (2.79) contradicts (2.78), hence (2.77) cannot hold. Theproof of the converse statement for the strictly concave case is similar tothe concave case and wll be omtted here.uTheorem 2.3 can be generalized to n-dimensional space along the linesof the remark preceding (2.32), as can be seen in the following theorem.Theorem 2.13 (Fenchel, 1951; Mangasarian, 1969). Let f be adifferentiable function on the open convex set C c R. It is concave if andonly if for every two points x' E C, x z E C

(x2 x') T [Vf(x2 ) Vf(x')] .c 0.2.80)It is strictly concave if and only if the inequa lity in (2.80) is strict for x' 96 x 2 .

Proof. Suppose that f is concave on C and let x' E C, x z E C. FromTheorem 2.12 we get

f(x2 ) - f(x') - (x2 - x' ) T VI(x ' )


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Concavity5Proposition 2.15 (Mangasarian, 1969). Let f be a concave functionand let A be a nonnegative number. Then F(x) = Af(x) is also a concavefunction. Let f, and f2 be concave functions. Then F(x) = f (x) + f2 (x) isalso concave.

The proof is straightforward. It follows from the above propositionthat every nonnegative linear combination of concave functions is alsoconcave.Denote by 0 the mapping from C c R" into D c R"', defined by

0(x) = (4 (x), ... , 4m(x)). (2.89)Proposition 2.16 (Mangasarian, 1969). Let , ... , m be concave(strictly concave) functions on C c R" and let f be a componentwisenondecreasing (increasing) and concave function on D c Rm. Supose thatD contains the range of 0, , ... , 4,. Then the composite function f4(x) _f(4 1 (x), ... , q5,(x)) is concave (strictly concave) on C.Proof. We prove only the strictly concave case. Since , , ... , 0, arestrictly concave, for every x' E C, x 2 E C, x' ^ x 2

c(Ax' + (1 A)x 2 )> A4(x') + (1 A)4(x2 ) ,2 . 9 0 )t h u s

f(Ax' + (1 A)x 2)> f(A4(x') + (1 A)(x 2 )) .2.91)By the concavity of ff(A4(x) +(1A4(x)) >_ Af4(x') + (1 A)f^(x z ).2.92)Hence4(Ax +(1A)x) > Af(x') + (1 A)f(x2 ) .2 . 9 3 )

n u

Proposition 2.17. Let f be a concave function defined on a convex setD c R' and let A be a given m x n matrix. Let C be a convex set in R"such that Ax E D for every x E C. Then f(Ax) is a concave function on C.


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3 6hapter 2Proof. The reader can easily verify that for every x' EC, x2 EC and0


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C o n c a v i t y7function that is not strongly concave is again the function f(x) = (x)4 onR. Let f(x) _ (x) 'a(x) z , where is some concave function defined onR, and a > 0. We can see that at the origin f'(0) = 0 = "(0) a, or"(0) = a > 0, contradicting that 0 is concave.We can now characterize strongly concave functions.

Proposition 2.19 (Diewert, Avriel, and Zang, 1981; Poljak, 1966;Rockafellar, 1976; Vial, 1982, 1983). Let f be a function defined on theconvex set C c RThen f is strongly concave if and only if there existsa positive number a such that for every x' E C, x2 E C, and 0


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38hapter 2Theorem 2.20 (Fenchel, 1951). Let f be a concave function defined ona convex set C c R. Then every local maximum of f at x* E C is a globalmaximum of f over all C.Proof. If x* E C is a local maximum off then there exists a positivenumber E such that

f(x*) >_ f(x)2.101)for all x E C satisfying Ix x < E. Let y be any point in C. It followsthat (1 A)x* + Ay E C for all 0 < _ A < _ 1, and for sufficiently small A > 01 1 ( 1 A ) )x* + A Y x *lI < e.2.102)Thus, f(x*) ^ f((1 A )x* + A y).2.103)From the concavity off it follows that

f(( 1 A ) )x* + A y) ^ (1 A )f(x * ) + A f(y).2.104)C ombining the last two inequa lities and dividing the result by A , we obtain

f(x* ) ^ f(y)2.105)Hence x* is the global maximum off over C.Clearly, the maximum of a concave function can be attained at morethan one point. The next result characterizes these points.Theorem 2.21 (Fenchel, 1951). The set of points at which a concavefunction f attains its maximum over C is a convex set.Proof. Let a* be the value of f at a maximizing point. Then, byCorollary 2.6, the upper level set

U(f, a*) = {x: x E C, f(x) ? a*}2.106)is convex.For strictly concave functions we have the following theorem.Theorem 2.22 (Fenchel, 1951). Let f be a strictly concave function,defined on the convex set C c R. If f attains its maximum at x* E C, thismaximizing point is unique.


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Concavity9The proof of this result is left for the reader. Note that the assumptionof strict concavity on all C can be weakened to strict concavity in aneighborhood of x*.Do concave functions always have a maximum? The answer to thisquestion is, of course, negative, as can be seen in the following example.Example 2.5. (a) Let C = {x: x ER, x >_ 1} and let f(x) = log x. Thisconcave function has no maximum since x is not bounded from above. Bychanging C to a compact (closed and bounded) set, however, this functionwill always have a maximum.(b) A compact domain for a concave function is not always sufficient

for having a maximum as demonstrated by the case of C ={x:xER,1sx


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4 0hapter 2T heorem 2 .2 4 (Fenchel, 1951). If f is a concave function defined onthe convex set C and attains its global minimum over C at an interior pointx of C, then f is constant on C.Proof. Suppose that f is not constant on C. Let y E C such that

f(y) > f(x ). Since x is an interior point, we can find a point z E C anda number 0 < A < 1 such thatx =,1y+(1A)z.2.111)By concavity w e havef(x ) > Af(y) + (1- A)f(z) > Af(x) + (1- A)f(x ) =f(x),2.112)

a contradiction.nDefinition 2.8. A point x belonging to a convex set C c R" is calledan extreme point of C if there are no points x' E C, X 2 EC, x' 0 x 2 , suchthat x = Ax' + (1 ,1)x 2 with 0 < A < 1.Now w e can state a result on minima of concave functions over compactsets.Proposition 2.25. If f is a continuous concave function on the compactconvex set C cR", then f attains a global minimum at an extreme pointof C.Proof. A complete proof of this proposition would require certainresults on convex sets that are not stated here, and, therefore, we present

here only an outline of the proof.Since f is continuous and C is compact, the minimum off on C isattained at some point x* E C. It can be shown that a compact convexset C cR" is equal to the convex hull of its extreme pointsthat is,the intersection of all convex sets containing the extreme points of C; see,for example, Roberts and Varberg (1973). If S is any subset of R", theconvex hull of S can be also obtained by taking the union of all convexcombinat ions of elements of S. If x', ... , xm are points in R ", the convexcombination of these points is given by A,x' + + A",X"where A,0, ... ,A m >_ 0 and A, + + A m = 1. It can be shown that if C cR" is aconvex set, x', ... , xm are elements of C, and if f is a concave functiondefined on C, thenf(A, x ' + . . . + AmX m ) > A If(X ' ) + . . . + A, (: m )2.113)


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Concavity1wherem

A i >0,...,Am ^O,1=1.2.114)Now, the classical Caratheodory's theorem states that every elementof the convex hull of S can be expressed as the convex combination of melements of S, where m min [f(v'),...f(vm)].(2116)But x* is a global minimum, so f must attain the value f(x*) at someextreme point v i E C .2.5. C oncave M athematical Programs

We continue here the discussion of extrema of concave functions byconsidering the following mathematical program:CPax f(x)2.117)subject to the constraintsg(x) ^ 0,= 1, ... , m,2 . 1 1 8 )h; (x) = 0,=1,...,p,2 . 1 1 9 )where f the g i , and the h;are, respectively, concave, convex, and affinefunctions on the nonempty open convex set C c R", containing the set ofpoints satisfying (2.118)-(2.119). It is assumed that the vectors of coefficientsa of the affine functions h(x) = (a) Tx h; are linearly independent.Program CP is called a concav e program .


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4 2hapter 2The set of points x E C satisfying (2.118) and (2.119) is called thefeasible set of CP and, by Corollary 2.6, it is a convex set. Thus CP is theproblem of finding the maximum of a concave function over a convex set.By Theorem 2.20 every local maximum of f at a point x* belonging to thefeasible set is also a global maximum of f over the whole feasible set. It isalso called a solution of CP. If the feasible set is nonempty, CP is called aconsistent program . If there exists a point z E C such that

g ; ( z) < 0,= 1, ... , m,2.120)h; ( x) = 0,= 1,... , p,2.121)

then CP is called strongly consistent . The strong consistency condition isalso known as Slater's constraint qualifications; see Slater (195 0). A generalresult on the consistency of a system of equations and inequalities is givenin the following proposition.Proposition 2.26 (Fan, Glicksberg, and Hoffman, 1957; Mangasarian,1969). Let 0,, ... , 4',, and 4, + ,, ... , 4 5 P be convex and affine functions,

respectively, on the nonempty convex set C c R. If the systemO( x) < 0,= 1, ... , m2.122); (x) = 0,= m + 1, ... , p2.123)has no solution x E C then there exist numbers a, , ... , ap , not all zero,such that a ; >_ 0 for i = 1, ... , m and for every X E CP a ; O ; (x) >_ 0.2.124)

Proof. Again, we present an outline of the proof. LetG(x)={y: yERP, y 1 > b 1 (x),i=1,...,m,y1 =.b,(x),i= m+1,...,p}

(2.125)and let

G = U G(x).2.126)XEC


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Concavity3By the assumptions, G does not contain the origin. This set is, however, anonempty convex set, for let y' E G, y z E G, thenAy; + (1 A)Y; > AOi(x') + (1 A)41(x 2 ) ^ Oi(Ax' + (1 A)x 2 ),

i = 1, ... , m2.127)Ay;+(1 , l )Y i =A (x')+(1 A)4i(x 2 )=O,(Ax'+( 1 A)x 2),i=m+1,...,p.2.128)Since G is a nonempty convex set not containing the origin, it is well knownfrom separation theorems for convex sets that there exists a hyperplaneseparating G and the origin. That is, there exists a set of points in R p, givenb y

H ={z: zE R",a Tz=0,a E R",a ^O}2.129)and called a hyperplane with the property that for every y c G we havea y >_ 0. Such a hyperplane is called a separating hyperplane. Varioustheorems on separating hyperplanes for convex sets can be found in theliterature; see, for example, Mangasari

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Generalized Concavity