Editors: M. Thoma and W. Wyner
I IPI
System Modelling and Optimization Proceedings of the 15th IFIP Conference Zurich, Switzerland, September 2-6, 1991
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Editor
Peter Kall Institute for Operations Research University o f Zurich Moussonstral3e 15 8044 Zurich Switzerland
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Preface
The 15th IFIP Conference on System Modelling and Optimization was held at the Uni- versity of Zurich, September 2 - 6, 1992. We had the pleasure to welcome about 260 par- ticipants; more than 200 contributed papers as well as 11 plenary lectures were presented. In the large variety of lectures all participants had plenty of opportunities to satisfy their personal interests, no matter whether they were more dkected e.g. to theoretical foun- dations of optimization, computational methods in mathematical programming, control problems, stochastic optimization or to modelling and optimization in applications.
Some of the authors had commitments to publish their results elsewhere, and others were not successful in passing the reviewing and selection process installed to cope with the standards and the available space. Nevertheless I believe that this proceedings volume reflects fairly well the outcome of the conference as well as the diversity of topics inten- sively discussed within IFIP TC 7 and its Working Groups.
Finally it is my pleasure to express my cordial thanks. Members of the International Pro- gram Committee I gave great support in soliciting papers for particular sections. Many members of the Local Organizing Committee t and of the International Program Commit- tee assumed the burden to meet here in order to select out of more than 400 contributions originally submitted those to be accepted for presentation and to structure the final pro- gram. Many experts gave their valuable support in the reviewing process for this volume. The cooperation with Springer-Verlag was smooth and emcient. And last but not least, the members of our Institute gave their support in preparing and running the conference, and in particular, without the immense effort of my secretary Mrs. G. Utzinger for all administrative matters I probably should have been lost!
Zurich, February 1992 Peter Kall
asee next page
INTERNATIONAL PROGRAM COMMITTEE
A.V. Balakrishnan, USA R.E. Burkard, A D. de Werra, CH/IFORS J. Dolezal, CS Y. Ermoliev, SU I.V. Evstigneev, SU E.G. Evtushenko, SU G. Feichtinger, A/OeGOR S. Flam,.N U. Haussmann, CDN J. Henry, F M. Iri, J P. Kall~ CH A. Kalliauer, A P. Kenderov, BG R. Kluge, D W. Krabs, D
A.B. Kurzhanski, A/SU I. Lasiecka, USA M. Lucertini, I K. Malanowski, PL M. Mansour, CH/SVI J. Mockus, SU M.J.D. Powell, GB A. Prekopa, USA A.H.G. Rinnooy Kan, NL S.M. Robinson, USA R.T.R.ockafellar, USA W.J. Runggaldier, I H. Schiltknecht, CH/SVOR. H.J. Sebastian, D J. Stoer, D P. Thoft-Christensen (chairman), DK J.P. Vial, CH
LOCAL ORGANIZING COMMITTEE
H. Amann A.D. Barbour K. Daniel D. de Werra K. Frauendoffer (secretary) H. Glavitsch H. Gr6flin
P. Kall (chairman) J. Kohlas M. Mansour H. Schiltknecht H.R. Schwarz P. St~hly J.P. ViM
Table of C o n t e n t s
I O p t i m a l i t y and D u a l i t y
Kummer B.
Dempe S.
Gessing R.
A Transformation for Solving a Discrete-Time Singular LQ Problem . . . . . . . . . . . 25
Gonz~.lez R.L.V. [ Tidball M.M.
Fast Solution of General Nonlinear Fixed Point Problems . . . . . . . . . . . . . . . . . . . . . 35
Peikert R. [ W6rtz D. / Monagan M. / de Groot C.
Packing Circles in a Square: A Review and New Results . . . . . . . . . . . . . . . . . . . . . . 45
'rammer C. / Tammer K.
with Linear Restrictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A Generalized Nonconvex Duality with Zero Gap and Applications ............ 65
II
II.1
M a t h e m a t i c a l P r o g r a m m i n g - A l g o r i t h m s -
Computational G e o m e t r y
Aurenhammer F. / St6ckl G.
Boissonnat J.D. / Devillers O. / Preparata F.P.
Computing the Union of 3-Colored Triangles .................................. 85
Viii
Pastitioning of Complex Scenes of Geometric Objects . . . . . . . . . . . . . . . . . . . . . . . . . 94
Roos T. ] Noltemeier H. Dynamic Voronoi Diagrams in Motion Planning: Combining Local
and Global Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Sugihara K. Application of the Delaunay Triangulation to Geometric Intersection Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Tada H. ] Shinoaki S. / Tonosaki T. ] Hyuga M. ] Nakai A.
Development and Implementation of the National Computer Mapping System (The Japanese Road Administration Information System} . . . . . . . . . . . . 122
I I . 2 D i s c r e t e O p t i m i z a t i o n
Arbib C. / Mocci U. / Scoglio C.
Methodological Aspects of Ring Network Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Br~sel H. ] Kleinau M.
On Number Problems for the Open Shop Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Dudzinski K. / Wahkiewicz S.
Fukao T. / Haxada T. / Wu J.
Continuous Modelling of Discrete Optimization Problems . . . . . . . . . . . . . . . . . . . . 165
Krause W. An Algorithm for the General Resource Constrained Scheduling
Problem by Using of Cutting Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Lassmann W. / Kogge R.
Nicoloso S. ] Nobili P.
A Set Covering Formulation of the Matrix Equipartition Problem . . . . . . . . . . . . 189
Ribeiro C. / El Baz D.
A Dual Method for Optimal Routing in Packet-Switched Networks . . . . . . . . . . . 199
IX
Tinhofer G. / Farnbacher E.
A New Lower Bound for the Makespan of a Single Machine Scheduling . . . . . . . 209
II.3 Linear P r o g r a m m i n g and Complementarity
Jlldice J.J. / Machado J. / Faustino A.M.
An Extension of Lemke's Method for the Solution of a Generalized" Linear Complementarity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Krivonozhko V.E.
Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
The Interior Point Method for LP on Parallel Computers . . . . . . . . . . . . . . . . . . . . 241
Roos C.
A Projective Variant of the Approximate Center Method for the Dual
Linear Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Numeric-Stability by All-Integer Simplexiterations . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
I I . 4 N o n l i n e a r P r o g r a m m i n g
Bulatov V.P. /Khamisov O.V.
The Branch and Bound Method with Cuts in E "+1 for Solving Concave
Programming Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Butnariu D. / Mehrez A.
On a Class of Generalized Gradient Methods for Solving Locally Lipschitz Feasibility Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Evtushenko Y.G. / Zhadan V.G.
Heredia F.J. / Nabona N.
X
Decomposition .............................................................. 311
Kl6tzler R.
Dmitruk A.V. Second Order Necessary and Sufficient Conditions of Pontryagin
Minimum for Singular Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
Joly-Blanchaxd G. / Quentin F. / Yvon J.P.
Optimal Control of Waves Generators in a Canal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
Klamka J.
Kocvara M. / Outrata J.V.
Problems with Variational Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
Krushev N.I. Nondifferentiable Design Optimization Involving the Eigenvahes of Control System Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
Kryazhimskii A.V.
Kurzhanski A.B. / Filippova T.F.
Maksimov V.L
Xl
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
Pickenhain S.
Roubicek T.
Sarychev A.V.
Pontryagin Extremals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
Tyatushkin A.I. ] Zholudev A.I. ] Erinehek N.M.
The Gradient Method for Solving Optimal Control Problems with
Phase Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
I I I . 2 D i s t r i b u t e d P a r a m e t e r S y s t e m s
Lagnese J.E. / Leugering G. / Schmidt E.J.P.G.
Modelling and Controllability of Networks of Thin Beams (Plenary Lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
Bello J.A. / Fern~ndez-Cara E. / Simon J.
Optimal Shape Design for Navier-Stokes Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
Casas E. ] Fern~.ndez L.A.
Choosing L ¢ Controls to Deal with Pointwise State Constraints . . . . . . . . . . . . . . 490
Duncan T.E. [ Maslowski B. [ Pasik-Duncan B.
On Boundary Control of Unknown Linear Stochastic Distributed Parameter Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
Flaudoli F. [ Tessitore M.
Kabzinski J.
XII
IV Stochastic Programming
Bouza Herrera C. Bounding the Expected Approximation Error in Stochastic Linear Programming with Complete Fixed Recourse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
de Groot C. / Wfirtz D. / Hanf M. / Hoffmann K.H. / Peikert R. / Koller Th. Stochastic Optimization - E~cient Algorithms to Solve Complex Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
Dupacova J.
On Interval Estimates for Optimal Value of Stochastic Programs . . . . . . . . . . . . . 556
Frauendorfer K.
On the Value of Perfect Information and Approximate Solutions in
Convex Stochastic Two-Stage Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564
Girko V.L. Integral Representation and Rezolvent Methods for Solving Linear
Stochastic Programming Problems of Large Dimension . . . . . . . . . . . . . . . . . . . . . . . 574
Kall P. / Mayer J.
Mulvey J.M. / Ruszczynski A.
Stochastic Programming Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598
Applied Modelling and Optimization
B i o l o g i c a l a n d M e d i c a l S y s t e m s
Kaddeche M. ] Bensaker B.
Radosch U.
An Econometric Analysis of the Need for Medical Care in Austria . . . . . . . . . . . . . 617
V.2 Computer Aided Modelling and D e s i g n
Bradley S.R. / Agogino A.M.
Caminada A. / Ousaalah C. / Giambiasi N. ] Colinas M.F. ] Kemeis J.
A Modelling Tool for Telecommunications Network Planning ................. 639
Koakutsu S. / Sugai Y. / Hirata H.
Block Placement by Improved Simulated Annealing Based on Genetic
Algorithm .................................................................. 648
Optimization Approach to the Modelling of Turbine Aircraft Engines ......... 667
Zakrzewski R.R. / Mohler R.R.
V.3 Ecology
tion Systems . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 689
Hirata H. Modelling of Flow Networks Using Information Coding: An Applica-
tion to Ecological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698
XlV
Abatement .................................................................. 706
Krechetov L.I.
Kypreos S.
Shaw R.W.
Development and Environmental Change .................................... 735
V .4 Economy and Energy
Krawczyk J.B. / Tolwinski B.
Management Problem ....................................................... 747
Decomposition Technique and Coordination of Optimal Energy
Production ................................................................. 757
Blondel H. / Moatti M.
Testing the Robustness of a New Decomposition and Coordination Algorithm with the Optimization of the French Nuclear Units
Maintenance Scheduling ..................................................... 767
Groscurth H.M. / K~mmel R.
Energy, Cost and Carbondioxide Optimization in Regional Energy
Systems with Periodic and Stochastic Demand Fluctuations .................. 787
Sannomiya N. / Akimoto K,
Modelling and Optimal Planning of a Gas Supply System for a
Power Plant ................................................................ 797
XV
F i n a n c i a l S e r v i c e s
Shreve S.E.
Martingales and the Theory of Capital Asset Pricing (Plenary Lecture) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 809
Berete I. [ Magendie 3. [ Moatti B.
Measuring the Position Risks on Capital Markets: A Mathematical
Programming Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824
3i D. [ Yin G.
On Diffusion Approximation of Some Term Structure Models . . . . . . . . . . . . . . . . 843
V . 6 P r o d u c t i o n a n d L o g i s t i c s
Agnetis A. / Signoretti F.
Minimization of Transfer Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855
Arbib C.
Tool Loading in FMS and VLSI Circuit Layout Problems . . . . . . . . . . . . . . . . . . . . 865
Bianco L. / Dell'Olmo P. / Speranza M.G.
A Decomposition Approach to a Scheduling Problem with
Multiple Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875
Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885
Mayr H.
Different Alternatives to Formulate the Robotics Collision Problem as an LP Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894
Shioyaa'aa T.
Production Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904
Dynamic Scheduling with Petri-Net Modelling and Constraint-Based Schedule Editing for Flexible Manufacturing Systems ........................ 913
V . 7 S t o c h a s t i c M o d e l l i n g
Aica~di M. / Di Febbraro A. / Minciardi It. Perturbation Analysis of Discrete Event Dynamic Systems Via Minimax Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925
Aicardi M. [ Miaciardi R. / Pesenti It. Minimizing the Customer Mean Flow Time in Simple Queue Networks: Upper and Lower Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934
Eschenaner H.A. / Vietor T. An Augmented Optimization Procedure for Stochastic Optimization and Its Application to Design with Advanced Materials . . . . . . . . . . . . . . . . . . . . . . 943
Jensen F.M. / Thoft-Christensen P. Application of Linear Decomposition Technique in Reliability-Based Structural Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953
Ramachandran V. / Sankaxanarayanan V.
Probability Measures of Fuzzy Events in Power Systems . . . . . . . . . . . . . . . . . . . . . 963
List of A u t h o r s
Agnetis A. 855 Agogino A.M. 629 Aicardi M. 925, 934 Akimoto K. 797 Andersen J.A. 241 Antila H. 757 Arbib C. 135, 865 Aurenhammer F. 77 Bello J.A. 481 Bensaker B. 611 Berete I. 824 Bianco L. 875 Blondel H. 767 Boissonnat J.D. 85 Bouza Herrera C. 541 Bradley S.R. 629 Br~sel H. 145 Bulatov V.P. 273 Butnariu D. 282 Caminada A. 639 Casas E. 490 Colinas M.F. 639 Colwell D.B. 833 de Groot C. 45, 546 DelrOlmo P. 875 Dempe S. 17 Devillers O. 85 Di Febbraro A. 925 Dmitruk A.V. 334 Dolezal J. 667 D5rfner P. 777 Dudzinski K. 155 Duncan T.E. 500 Dupacova J. 556 El Baz D. 199 Elliott R..J. 833 Erinchek N.M. 456 Eschenauer H.A. 943 Evtushenko Y.G. 292 Farnbachet E. 209
Faustino A.M. 221 Ferngndez L.A. 490 Fern~ndez-Cara E. 481 Filippova T.F. 394 Flandoli F. 510 ~auendorfer K. 564 Fukao T. 165 Fukuda T. 913 F~tl6p J. 777 Gessing R. 25 Giambiasi N. 639 Girko V.L. 574 Gonz~lez R.L.V. 35, 885 Groscurth H.M. 787 Haasis H.D. 689 Hanf M. 546 Harada T. 165 Heredia F.J. 301 Hirata H. 648, 698 Hoffer J. 777 Hoffmann K.H. 546 Holnicki P. 706 Hyuga M. 122 Jensen F.M. 953 Ji D. 843 Joly-Blanchard G. 344 Jddice J.J. 221 Kabzinski J. 520 Kaddeche M. 611 Kall P. 58O Kaluszko A. 706 Kernels J. 639 Khamisov O.V. 273 Klamka J. 354 Kleinau M. 145 KlStzler R.. 323 Koakutsu S. 648 Kocvara M. 364 Koller Th. 546 Krause W. 175
Krawczyk J.B. 747 Krechetov L.L 716 Krivonozhko V.E. 231 Krushev N.I. 374 Kryazhimskii A.V. 384 K~mmel R, 787 Kummer B. 3 Kuru S. 657 Kurzhanski A.B. 394 Kypreos S. 725 Lagnese J.E. 467 Lahdelma tL 757 Lasiecka I. 530 Lassmann W. 185 Lautala P. 757 Leugeti'ng G. 467 Levkovitz R. 241 Machado J. 221 Magendie J. 824 Maksimov V.I. 404 Maslowski B. 500 Matousek O. 667 Mayer J. 580 Mayr H. 894 Mehrez A. 282 Minciaxdi R. 925, 934 Mitra G. 241 Moatti B. 824 Moatti M. 767 Mocci U. 135 Mohler R.l:t. 677 Monagan M. 45 Mori K. 913 Mulvey J.M. 588 Muramatsu K. 885 Myslinski A. 414 Nabona N. 301, 311 Nakai A. 122 Nicoloso S. 189 Nohili P. 189
XVIII
Noltemeier H. 94, 102 Oussalah C. 639 Outrata J.V. 364 Pasik-Duncan B. 500 Peikert P~ 45, 546 Pesenti It. 934 Pickenhain S. 424 Preparata F.P. 85 Quentin F. 344 Radosch U. 617 Ramachandran V. 963 Ribeiro C. 199 Rofman E. 885 Rogge R. 185 Boos C. 251 Roos T. 94, 102 Roubicek T. 433 Ruszczynski A. 588 Ruuth S. 757 Sankaranarayanan V. 963 Saanomiya N. 797 Saxychev A.V. 440 Say A.C.C. 657 Schindler Z. 667 Schmidt E.J.P.G. 467 Schneider W. 261 Scoglio C. 135 Sen S. 598 Shaw R.W. 735 Shinoaki S. 122 Shioyama T. 904 Shteve S.E. 809 Signoretti F. 855 Simon J. 481 Spengler T. 689 Speranza M.G. 875 St~ickl G. 77 Sugai Y, 648 Sugihara K. 112 Tada H. 122
'rammer C. 55 Tammet K. 55 Tessitore M. 510 Thach P.T. 65 Thoft-Christensen P. 953 Tidball M.M. 35 Tinhofer G. 209 Tolwinski B. 747 Tonosaki T. 122 Triggiani R. 530 Trigub M.V. 449 Tsukiyama M. 913
XIX
Tyatushkin A.I. 456 Verdejo J.M. 311 Vietor T. 943 Walukiewicz S. 155 Wu J. 165 W{irtz D. 45, 546 Yin G. 843 Yvon J.P. 344 Zakrzewski R.R. 677 Zhadan V.G. 292. Zholudev A.I. 456 Zirkelbach C. 94
3
with Applications to Optimization Problems
Bernd Kummer
Unter den Linden 6, D-0-I086 Berlin
PF 1297
KEY WORDS
od, Generalized derivatives, Multifunctions, Convergence analy-
sis, Approximate solutions, Critical points in optimization.
I. INTRODUCTION
be written in terms of an equation
(i) F(x) = 0; F: R p--~ R q locally Lipschitz.
Typical examples of such functions are those which include pro-
jection maps arising from complementarity problems or from opti-
mality conditions in standard optimization models (see e.g. Refs.
[6], [8], [9], [I0], [18]). In this paper we study two well known
problems that are completely solved in the cl-case.
(PI) Under which conditions the inverse of F (put p = q ) and,
more general, the implicit function X = X(a, t) concerned
with the parametrized equation
(l,a,t) F(x, t) = a ; F: R n + m .~ Rn loc. Lipschitz
are locally single-valued and Lipschitz ?
(P2) Under which conditions equation (i) can be solved by some
Newton-type method ?
tions in wide generality including also pseudo-Lipschitzian
behaviour of the inverse as a multifunction ([i] - [5], [7],
[13], [14], [16], [17], [25]); for some survey and more refer-
ences see [2] and [25].
Intensive studies devoted to (P2} began few years ago ( Refs.
[6], [ii], [12], [15], [19] - [24]) and led to rather differently
restrictive concepts to approximate the equation
(2) F(x + u) = 0 ( x given, u searched )
by certain generalized derivatives of F at x. The related auxil-
iary problems are linear complementarity problems, in most cases.
To obtain some unified approach for both problems, we assign, to
F, any multifunction GDF defined on R p+p with non-empty images
GDF(x, u) C R q satisfying GDF(x, 0) = [ 0 ]. We denote the
class of such multifunctions by M and consider GDF(x, u) as
some generalized directional derivative of F at x in direction u.
In § 2 we show how problem (PI) can be solved by some necessary
and sufficient condition in terms of derivatives GDF o£ the kind
GDF(x, u) := TF(x, u) := [ v ¢ R q / v = lim [ F(x'+ tu)-F(x'} ]/t
for some sequences x' --~ x and t ~ 0 ].
The map TF denotes the limit sets DF(X,U) introduced and studied
in [26] and [27]. D@e to finite dimension and Lipschitz property
these sets are always nonempty.
Concerning (P2) we study (§ 3) the abstract algorithm ALG(u)
defined by
!3,~) Find u such that [F(x) + GDF(x, u)] 6~ ~ | F(x} | B +
put x := x + u and proceed.
Here, ~ is a nonnegative fixed parameter, the term ~| F(x)| B
denotes the closed ball with centre O and radius ~| F(x) ~ , and
algebraic set-valued operations indicate the set of elementwise
results , e.g. y + X = { y + x / x ¢ X |.
Obviously, ~ describes the error tolerated when solving the
system
(3,0) -F(x) ¢ GDF(x, u)°
We shall demonstrate that ALG(~) makes sense (see Def.l ) if two
conditions, called (CI) and (CA) are satisfied. They require
uniform (w.r.t. x) injectivity of GDF(x, .) as well as some
approximation property for x near to some zero x* of F. More-
over, if F is a Lipschitzian homeomorphism of R n into itself then
they turn out to be even necessary for relevant settings of GDF
namely for GDF = TF and ( if usual directional derivatives
F'(x, u) exist ) for GDF = F'
Generally, our approach helps to clarify how the function F, its
zero x* and the map GDF must be connected in order to ensure
that ALG(~) may be applied for sufficiently small positive ~. In
particular, we identify GDF with the maps TF, F' as well as with
the multifunctions CF and CLF that we are defining now.
CF(x,u) := [ v / v = lim ( F(x + tu) - F(x) )/t for some
sequence t ~ O l,
the set of all directional limits which form the contingent
derivatives in the sense of Ref. [i].
CLF(x,u) := 6F(x)u := [ v / v = Au for some A z 8F(x) I
where 8F(x) stands for the generalized Jacobian of F at x , see
[4], [5].
Finally, for pcl-functions F ( i.e. F is continuous and fulfils
F(x) = FU(x) (x) where F U(x) is one of finitely many cl-func -
tions ) we also regard
PDF(x, u) := { DF~(x) u / FU(x) = F(x) i, see Ref. [ii].
Note that
(4) CF C TF ~ CLF and cony TF(x, u} = CLF(x, u);
If F is PC 1 then CF = F' and CLF C PF, but TF + CLF
is possible (example in [13]).
In § 4 we shall discusse the meaning of our hypotheses for the
example of Kojima's function F whose zeros characterize Karush-
Kuhn-Tucker points of optimization problems.
2. AN IMPLICIT FUNCTION THEOREM
We call the locally Lipschitz F in (l,a,t} regular at (x*,t*,a*)
satisfying F(x*,t*) = a* if there are neighbourhoods N(x*) and
N(a*,t*) for x* and (a*,t*), respectively, such that (l,a,t) has
6
exactly one solution X(a,t) in N(x*) whenever (a,t) z N(a*,t*)
and if, additionally, X is Lipschitz on N(a*,t*).
Theorem 1 F is regular at (x*,t*,a*) if and only if
(5) 0 ( TF( (x*,t*), (u , 0) ) for all u z a n t 0 /
Moreover, if (5) is true then there holds the equivalence
(6) u E TX( (a*,t*), (=, ~) ) iff = ~ TF( (x*,t*), (u, ~) ).
This theorem (Proof in [14]) has some consequences.
Assume F does not depend on t. Then Th. 1 strengthens Clarke's
well-known inverse function theorem since TF ~ CLF and strict
inclusion is really possible. Since TF is closed, uniformly
bounded and homogeneous in u, Cond. (5) then means that the
multifunction TF( x* , .) is uniformly (w.r.t. x near x*) injec-
tive, i.e. there are some c > 0 and a neighbourhood Q of x* such
that
(7) c ~ u | ~ ~ v i for all v ~ TF(x, u) and x ~ Q.
Formula (6) now particularly says that (7) implies surjectivity
of TF(x, .) for all x E Q. This fact is of certain importance
when GDF = TF in (3,~).
If, more general, TF can be written as the sum of the (canonical-
ly defined) partial derivatives
then (5) means
(8) 0 / TxF((x*,t*), u) for all u ¢ R n \{0l
which is equivalent to the requirement that the inverse of
F(. ,t*) , say • , is locally well-defined and Lipschitz ( near
(x*,a*) ). One then obtains
(9) TX( (a*,t*), (~, ~) ) = T~( a* , ~ - TtF((x*,t*), 5) ),
i.e. the set-valued version of the known C I- formula for the
derivative of X. Though (7) is generally violated it holds for
important particular cases. Among them the next one can be ap-
plied to Kojima's function [9] for optimization problems (Proof
in [13]).
(ii) the partial derivatives DyH(.,.) w.r.t, the second variable
of H exist, and DyH , H and V are locally Lipschitz,
(iii) If v c TV( t, ~) and s k ~ 0 ( k --~ - ) then there are
t k -~ t such that v is a cluster point of the sequence
v k = ( V( t k + s k ~) - V(t k ) ) / s k .
Then TF((x*,t*), (u, ~) = TxH((x*,t*), u)
+ DyH{x*, V(t*)) ( TV( t*., ~) ) .
Condition (iii) holds e.g. for Lipschitzian functionals V and for
the complementarity map V(t) := C(t) := (t +, t-) where
ti + = max [0, t i} and t i- = min (0, ti}.
In both cases it holds V'(t, 0) (: TV(t, 0) = CLV(t, ~).
3. NEWTON-type ME~"HODS: Local Convergence
3.1. Conditions for Convergence
lim o(t)/t = 0 for t W 0.
We investigate the iterative process ALG(e) given by (3,~). The
results of this chapter are verified in [15]. Let x* be some zero
of F in (i) and GDF z M.
Def.l The triple (F, x*, GDF) is called feasible if, for each
positive E, there are positive ~ and r such that ALG(~) generates
an infinite sequence x k satisfying
I x k + l - x * 0 ~ ~ 1 x k - x * I whenever ~ x 0- X * n ~ r .
Note that this definition implies solvability of each subproblem
(3,~) with x = x k.
Lemma 2 The triple (F, x*, GDF) is feasible if there exist some
neighbourhood Q of x* , c ) 0 and some function o(.) such that
the conditions
(CI) c ~ u ~ ~ inf [ ~ v ~ / v z GDF(x, u) ] and
(CA) F(x)+ GDF( X, u) C GDF(X, u + X - X*) + o( ~ X - X* ~ ] B
hold for all x z E and u ~ X.
Moreover, if both conditions are satisfied and L is some Lip-
schitz constant for F on E then, given any z ~ I, the reals u and
r in Def.l can be taken as follows:
u = c s /(2 L), r sufficiently small such that x* + rB C E
and o(s)/s ~ rain { c ~/ 2 , c u~ 2 ] for 0 ~ s < r.
The proof of the Lemma is straightforward and shows, in particu-
lar, that x* satisfies
F(x) | ~ (C/2) I x - X* | if | X - X* | ~ r.
Remark 1 If one determines solutions of the subproblems (~0) the
convergence is of order o(.) appearing in (CA):
c I x k + l - x ~ I ~ o~ I x k - x * I ~ •
Having the C I- case in mind [18] there is no question about the
need of (CI) for our purpose.
On the other hand, condition (CA) appears something artificial.
In fact, for the moment regard it as a technical assumption.
However, (CA) can be simplified and made tractable for each of
the maps GDF specified in the introduction.
Theorem 2 Let GDF be one of the mappings CF, TF, CLF or PF.
Then Cond. (CA) is equivalent to
(CA*) F(x) + GDF( x, x* - x) (~ o( ~ x - x* | ) B.
Note that PF is defined only for PC 1 functions, the other map-
pings for arbitrary locally Lipschitz functions. The theorem
permits to put u = x* - x in (CA).
In (CA) and (CA*) , the o(.)-functions may differ from each
other. Specifically, denoting the latter by o'(.} the relation
between them is very simple: o(.) is the upper semicontinuous
closure of o'(.) defined by
o(t) = lim sup o'(s). Hence lim sup o(t)/o'(t) is finite.
s -~ t t ~ 0
Condition (CA*) has a simple interpretation when using that GDF
9
is positively homogeneous in u. Setting x = x* + t u ( t > 0 ,
| u ~ = 1 ) the quotients q(u, t) = (F(x*) - F(x* + tu) ) / t
and any elements v(u, t) belonging to GDF(x* + tu, - u) fulfil
lim sup | v(u, t) - q(u, t) J = 0 uniformly w.r.t, u.
t ~ 0.
This immediately requires diam GDF(x* + tu, -u) -~ 0 as t ~ 0,
which is a strong condition in the class of all Lipschitz func-
tions.
If GDF ~ CF the proof of Th. 2 is straightforward d~e ta the fact
that GDF is sublinear in u w.r.t, inclusions, i.e.
GDF(x, ul + u2) CZ GDF(x, ul) + GDF(x, u2). We may then write
F(x) + GDF(x, u ) C F(x) + GDF(x, x*- x ) + GDF(x, u + x - x* )
(:~ o( | x - x* J ) B + GDF(x, u + x,- x* ).
However, if GDF = CF then much more investigations are needed
(even if F has directional derivatives F').
For the following two characteristic examDles, namely pcl-func -
tions and point-based approximations, Cond. (CA*) is always
satisfied.
P~-functions F
Cond. (CA*) holds true for every mapping GDF considered in Theo-
rem 2. Indeed, since F is continuous the active index sets
I(x) = t U / FU(x) = F(x) 1 are finite and included in I(x*)
for x near x*. Since all F ~ are C 1 we also have
F~(x) + DF~(x] (x * - x) ~ oU(|x * - x |) B.
Therefore, setting o(.) = max o~(.) we already see that PF ful-
fils (CA*). The same statement for the remaining mappings GDF
follows from (4).
Point based approximations (PBA)
Let us suppose that F has directional derivatives F'(x, u) which
define a PBA in the sense of Ref. [24]. This means in our set-
tings: There is some positive K such that
(i) ~ F(x + u) - F(x) - F'(x,u) ~ ~ K | u |2 /2 and
(ii) The functions r = r(z) := F'(x, z-x) - F'(x', z-x') are
Lipschitz near x* with the constant L = K ~ x - x' ~.
10
Using only (i) we observe that, for any zero x* of F and for
u = x* - x the inclusion
0 = F(x+u) e F(x) + F'(x, u) + K (| u |2 12) B
is valid giving (CA*) for GDF = F'
Cond. (CA*) restricts the class of locally Lipschitz functions
which allow to use ALG(~) for computing some zero x*. In particu-
lar, there is no reason to assume that, as in the C I- case,
ALG(~) may be applied (with some appropriate map GDF 9 whenever
F is Lipschitzian invertible.
The feasibility of some triple (F, x*, GDF) obviously depends on
GDF. The larger GDF(x,u) the stronger (CI) and (CA) are. There-
fore it makes sense to ask whether these conditions are even
necessary for feasibility. We are going now to deal with this
question.
In this section we assume that
(i0) p = q in (I) and F is regular at (x* , 0) (see § 2).
Then F establishes a Lipschitzian homeomorphism between some
neighbourhoods Q(x*) and Q(0). We denote the inverse function
doing from ~(0) onto ~(x*) by ~.
Lemma 2 Let GDF be one of the mappings CF, TF, CLF or PF,
suppose (10) and let x z Q(x*). Then (3,0) has at least one solu-
tion u. The same is true for directionally differentiable F and
GDF = F'
The key for proving this Lemma consists in the equivalence
(ii) v ~ GDF(x, u) iff u ¢ GD#(F(x), v)
which implies that GDF(x, .) is surjective. Because of (i0),
formula (ii) holds for CF (see e.g. [i]) and TF (6). Concerning
GDF = F' the existence of directional derivatives ~' as well as
(ii) are shown in [15].
Although (ii) fails to hold for CLF and PF, the Lemma now follows
from (4). Formula (II) and Th. 1 play also "a crucial role for
proving the next proposition.
11
Theorem 3 Let (F, x*, GDF) be feasible, and suppose (10).
If GDF = CLF then (CA) is valid.
If GDF = TF or if F is directionally differentiable and GDF = F'
then (CA) and (CI) are true.
Note that the theorem characterizes a nonempty subclass of direc-
tionally differentiable and Lipschitzian homeomorphisms F such
that (F, x*, F') is never feasible: F belongs to this class iff
its directional derivatives violate (CA*). Since F'(x, u} is
contained in the sets TF(x, u) and CLF(x, u), Cond. (CA*) is
violated also for these generalized derivatives.
In [12] one finds a pathological example where ALG(~) with ~= 0
and GDF = F' generates an alternating sequence whenever the
initial point differs from x*.
3.3. Mappings in normed spaces
We now suppose that
F: X --) Y is locally Lipschitz, X Banach space, Y normed space.
Then the discussed realizations of GDF(x, u) may become empty or
are not defined (CLF).
Nevertheless, if GDF z M ( now defined in a corresponding way )
then Lemma 2 remains true; and Theorem 2 still holds for GDF = CF
if, additionally, each sequence of the kind
( F(x + tkU) - F(x) )/ t k , t k ~ 0
has some accumulation point in Y. The next Kantorovich-type theo-
rem makes essentially use of these facts.
Theorem 4 Let z E X and c, ~ and ~ be positive.
Suppose that F is Lipschitz with constant L on z + dB and, for
all x in z + 6B,
(i) F has directional derivatives F'(x, u) ,
(ii) ~ By C cl F'( x, B X) ,
(iii) R F(x + u) - F(x) - F'(x, u)R ~ o( ~ u ~ ) ,
(iv) 2c ~ u ~ ~ ~ F'(X, U)
12
Then after setting x 0 = z, ALG(u) determines a zero of F within
z + (6/2) B whenever u and ~ F(z) ~ are positive and sufficient-
ly small. The related constants are given below.
For simplicity, suppose that o(.) is upper semicontinuous, o(t)It
converges monotonously, and c < L .
Fix some z < 1 and put ~ = c s / L. Now take r such that
r < 5/2 and o(r)/r < ~ c.
Finally, impose the condition | F(z) ~ < ~ r on z.
Remark 2 With the given constants, ALG(a) determines the unique
zero of F in the ball z + rB.
Our Th. 4 generalizes Th. 3.2 in Ref. [24] not only by the fact
that Y is any linear normed space. As the main differences we
note the here included error parameter a and the absence of the
assumption
| F'(z, ul) - F'(z, u2) ~ ~ q ~ ul - u2 ~ with some q ) 0
being a key of Robinson's proof. On the other hand we have to
impose the uniform surjectivity (ii) and injectivity (iv) of F'
which follow under the assumptions in [24].
4. PRIMAL-DUAL SOLUTIONS IN OPTIMIZATION PROBLEMS
We consider the usual finite-dimensional optimization model
P(a,b,c) minimize f(x) - (a, x) s.t. g(x) ~ b and h(x) = c
with parameters a, b, c where f, g and h have locally Lipschitz
first derivatives (or, equivalently, are CI'I). Put s = (x,y,z)
where y and z are the dual vectors belonging to g and h, respec-
tively. One easily sees that the Karush-Kuhn-Tucker points of
P(a,b,c) may be identified with the solutions of the system
{12) F(s) = (a,b,c) T
+D F Df(x) + y g(x) + z Dh(x)
F(s) = ~g(x) - y-
~h (x)
Let F(s*) = 0. If F is regular at (s*, 0), Th. 1 presents the
T-derivatives of the inverse ~.
13
To determine the sets TF( (x*, y*, z*), (u,v,w) ) or briefly
TF( s*, o), we note that F = Q * V where Q is the matrix
Qlx) = g(x) T 0 -E
h(x) T 0 0
and V denotes the column-vector V(y,z) = (i, C(y),z) T including
the complementarity map C(.) of $2. Here g and h are row-vec-
tors, the derivatives are column-vectors. The Lagrangian is then
nothing but the scalar product L(s) = ( (f, g, 0, h~, V(y,z) ),
and our main tool is the mapping H(u) = T(DxL(., y*, z*)(x*, u).
Applying Lemma 1 one finds that
(~, ~, ~) ~ TF(s*, o) if and only if
E H(u) + ~ Pi Dgi(x) + ~ wj Dhj(x)
(13) ~i = (Dgi(x) , u ) - qi
~j = (Dhj(x) , u )
The latter means
p + q = v , pi qi ~ 0 , (14) Pi = 0 if Yi* < 0 and qi = 0 if Yi* > 0.
Our regularity condition (5} claims that (13), for (a,~,~) = O,
implies (u,v,w) = 0 . As shown in [13] this is equivalent to the
linear independence constraints qualification (LICQ) and certain
non-singularity of H(.) w.r.t, the (large) tangent space of the
feasible set.
For C2-functions, this regularity condition can be derived by
several approaches ([8] [i0], [25]), since H(u) consists of
D2xL(S*)U only, TF = CLF, and system (13) (as well as (3,0) ) is
of linear complementarity type.
Let us return to C I'I
If (5) is not true then (CI) is violated for GDF = TF and GDF =
CLF, but CF could satisfy (CI) and (CA); take x* = 0 for
min sign(x) * x 2 + x 3 without constraints. The related func-
tion F(x) = 2 ABS(x) + 3 x 2 has the property that all points z
sufficiently close to x* satisfy the assumptions of Th. 4 (with
certain c, 0, ~) except z = x* where F'(x*, .) is not surjective.
14
However, in such situations some problems may arise from a large
error when CF or directional derivatives are determined by dif-
ference approximations.
In what follows let regularity hold.
Condition (CI) is now satisfied for TF and CF. If Df, Dg and Dh
are directionally differentiable so are F and ~, ~' can be
obtained via (ii) and (CI) holds also for GDF = F' (§ 3.2).
For the related critical-value function
m(a,b,c) = f(x) - (a, x) where (x, y, z) = • (a,b,c),
the first derivative Dm(a,b,c) = - (x, y+, z) and the (l.l)-
derivative T(Dm) ( (a,b,c), (~,~,z) ) are now given via (13); in
[14] one finds explicit formulae including also the case that f,
g and h depend (smoothly enough to apply Lemma I) on some further
parameter. In this context it may be worth noting that, if y*
0, the map Dm is regular at ((0,0,0),-(x*,y*+,z*)) iff y* fulfils
the strict complementarity condition.
Nevertheless, (CA) may fail to hold, in general.
On the other hand, it is trivially satisfied whenever Dr, Dg and
Dh are pcl-functions.
This case is of some importance for two-level optimization where
the objective of the higher level (master's problem) is some
critical-value function m(.) of the lower one. Since F is P~ ,
the locally Lipschitz inverse • (near 0 = F(s*) ) as well as
the derivatives Dm(.) are again of pcl-type.
This becomes clear when defining the locally inverse functions
• ~ := (FU) -I for such ~ that DF ~ is regular on some neighbourhood
of s*.
New York, 1984
Birkha6ser, Basel, 1990
[3] Aze, D. An inversion theorem for set-valued maps. Bull.
Austral. Math. Soc. 37 (1988) pp. 411-414
[4] Clarke, F.H. On the inverse function theorem. Pacific Journ.
Math. 64, No. 1 (1976) pp. 97-102
15
NewYork, 1983
inequality and nonlinear complementarity problems: A survey of
theory, algorithms and applications. Mathematical Programming 48,
(1990) pp. 161-220
pp. 1-56
ming Study 21, (1984) pp. 127-138
[9] Jongen, H.Th., Klatte, D., Tammer, K. Implicit functions and
sensitivity of stationary points. Preprint No. i, Lehrstuhl C f~r
Mathematik, RWTH Aachen, D-5100 Aachen
[i0] Kojima, M. Strongly stable stationary solutions in nonlin-
ear programs. In: Analysis and Computation of Fixed Points, S.M.
Robinson ed., Academic Press, New York, 1980 pp.93-138
[ii] Kojima, M. & Shindo, S. Extensions of Newton and quasi-
Newton methods to systems of P~ equations. Journ. of Operations
Research Soc. of Japan 29 (1987) pp. 352-374
[12] Kummer, B. Newton's method for non-differentiable func-
tions. In: Advances in Math. Optimization , J.Guddat et al. eds.
Akademie Verlag Berlin, Ser. Mathem. Res. Vol 45, 1988 pp.i14-125
[13] Kummer, B. Lipschitzian inverse functions, directional
derivatives and application in cl'l-optimization. Working Paper
WP- 89-084 (1989) IIASA Laxenburg, Austria
[14] Kummer, B. An implicit function theorem for ~'l-equations
and parametric ~'l-optimization. Journ. Math. Analysis & Appl.
(1991) Vol, 158, No.l, pp.35-46
[15] Kummer, B. Newton's method based on generalized deriva-
tives for nonsmooth functions: convergence analysis. Preprint
(1991) Humboldt-University, Deptm. of Appl. Mathematics
[16] Langenbach, A. Ober lipschitzstetige implizite Funktionen.
Zeitschr.f6r Analysis und ihre Anwendungen Bd. 8 (3), (1989) pp.
2 8 9 - 2 9 2
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nonsmooth optimization. Preprint (1991), Deptm. of Mathem.,
Wayne State University, Detroit, Michigan 48202, USA
[18] Ortega J.M. & Rheinboldt W.C. Iterative Solution of Nonlin-
ear Equations of Several Variables. Academic Press, San Diego,
1970
[20] Pang, J.-S. & Gabriel, 8.A. NE/SQP: A robust algorithm for
the nonlinear complementarity problem. Working Paper, (1991),
Department of Math. Sc., The Johns Hopkins Univ., Baltimore
Maryland 21218
Ph.D.Dissertation, (1989), Department of Industrial Engineering,
Univ. of Wisconsin-Madison
[22] Qi, L. Convergence analysis of some algorithms for solving
nonsmooth equations. Manuscript, (1991), School of Math., The
Univ. of New South Wales, Kensington, New South Wales
[23] Ralph,D. Global convergence of damped Newton's method for
nonsmooth equations, via the path search. Techn. Report TR 90-
1181, (1990), Department of Computer Sc., Cornell Univ. Ithaca,
New York
[24] Robinson, S.M. Newton's method for a class of nonsmooth
functions. Working Paper, (1988), Univ. of Wisconsin-Madison,
Department of Industrial Engineering, Madison, WI 53706
[25] Robinson, S.M. An implicit-function theorem for a class of
nonsmooth functions. Mathematics of OR, Vol. 16, No. 2, (1991)
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rials of Lipschitz vector-valued functions. Nonlinear Analysis
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Optimality Conditions for Bilevel Programming Problems
S. D e m p e
F a c h b e r e i c h M a t h e m a t i k , T e c h n i c a l U n i v e r s i t y C h e m n i t z ,
P S F 964, O - 9 0 1 0 C h e m n i t z , F R G
1 I n t r o d u c t i o n
Consider a practical situation in which two decision makers try to realize maximal values of different objective functions. Assume that, while controlling their own variables, the decision makers are forced t o act within a fixed hierarchy. That means that the first decision maker or leader is asked to choose his strategy !/0 first, thereby influencing the objective function as well as the admissible set of the second decision maker or follower. Then, after communicating the value y0 to the follower, the latter selects his decision x ° = x(y °) which thus may be considered as the follower's optimal reaction on the leader's choice. In order to avoid certain difficulties which are due to nonunique optimal solutions in the follower's problem (el. e. g. [7]), throughout the paper, we assume the existence of a uniquely defined function x(.) describing the optimal reactions of the follower on the leader's choices. Since the leader's objective function depends on the follower's strategy too, only now, after announcement of z °, the leader is able to evaluate the real value of his choice. The problem studied here is the leader's problem: How to select a strategy y0 such that it is an optimal one ? Now, let fo, ho : / ~ x R = ~ R denote the objective functions of the leader and the follower, resp., and let the admissible sets of both decision makers be given by the help of the functions f : R '~ --+ R ~, h : R ~ × R '~ --* R q. Then, the optimal reaction of the follower on the leader's choice is defined as the optimal solution of the following parametric optimization problem:
x(y) e ~l(y) := Argmin{ho(x,y)lh(x,y ) < 0}. (1)
The leader's problem can be stated as follows:
min{fo(x,y)lf(y) < O,x E ~(y)}. (2) Y
Note that this problem is well-defined since problems (1) are assumed to have unique optimal solutions x(y) for all y. By use of the implicitly defined function x(.), problem
18
(2) is allowed to be transformed into the (nondifferentiable and nonconvex) mathematical programming problem
min{g(y)l/CY ) ___ O} (3)
the implicitly defined objective function of which is given by
g(y) := fo(xCy), y). (4)
Now, all results concerning optimality conditions for nondifferentiable optimization pro- blems could be applied. But, due to the relations of (I), (2) to (3), each differential calculus applied to the function g has to be translated into a calculus applied to the implicitly defined vector function xC. ) and, then, conditions are to be established guaran- teeing applicability of this calculus to x(.). Promising attempts for describing optimality conditions for (3) are e. g. based on an approach using a differential calculus motiva- ted by certain cone-approximatlons of the epigraph of the functions involved (cf. e. g. [12]). Here, we use the contingent cone approximating the epigraph of g, resulting in the contingent derivative of the function g (cf. [12]). But, for existence of this derivative we need at least upper Lipschitz-continuity of the function g(-) and, by definition, also upper Lipschitz-continuity of the vector-valued function x(.). Well-known results gua- ranteeing this property have been obtained for problems (1) involving sufficiently smooth functions [9]. Thus, whether this is not unavoidable, it is convenient for us to assume also f0, f being sufficiently smooth. Since the function x(.) is vector-valued, we use a set-valued version of the contingent derivative [1, ch. 7] for the function z(.). Then, using a method for computing this derivative presented in [3], we are able to describe the contingent derivative of the function g(.). This derivative is then used to state two combinatorial optimization problems whose optimal values are used for verificating the optimality conditions.
2 The contingent derivative of the optimal solution of the lower level problem
Consider the lower level problem (1) at the point y = y0 E R "~ and let x ° E k~(y °) C_ R". Assume that the functions ho, h are sufficiently smooth and convex with respect to x for each y in a certain open neighbourhood of y0. Thus, (1) is a convex, parametric optimization problem. Let hi(., .) denote the components of the vector-valued function h(-). In what follows, we need two different regularity conditions. The first is the well-known Slater's condition: (A1) {xlh(x,y °) < 0} # 0. The second one is a slight modification of the constant rank condition investigated e.g. in [5], [81:
19
(A2) For each 0 ~ K C_ I(x °, yO) := (j]hj(xo, yO) = 0} the Jacobian matrices
(v~h,(~(y), y)li e K)
are of constant rank in some open neighbourhood B(y °) of y0. It is well-known that, for y E B(y °) and if (A1) is satisfied, then the set of KKT-multiplier vectors
U(x,y) := {u > Olu~h(x,y) = O, V=L(x,y,u) = O)
(where L(., .,.) dcnotes the usual Lagrangian of problem (1)) is nonempty for a certain vector x satisfying h(x,y) <_ 0 iff x e @(y). Then, U(x,y) is a bounded polyhedron, i.e., it is equal to the convex hull of the finite set EU(x, y) of its vertices. Moreover, if (A2) is also satisfied, then for each sequence {Y~}~I converging to yO, the vertices of U(x °, yO) are the only cluster points of an arbitrary sequence t ~ t {u },=,,u 6 EU(z(y') ,y ') , t = 1 ,2 , . . . [2]. This property will be used implicitly in what follows. Now, the function x(.) : B(y °) ---, R ~ satisfying {x(y)} = q~(y) for each y in the sufficiently small neighbourhood B(y °) of yO whose application has been announced in the introduction exists provided that the following strong second-order sufficient optimality condition is satisfied [6]: (A3) For each triple (x, d, u) with
e ~(N°),u e u(~,N°),
Vxhi(x,y°)d = 0, Vi e g+(u) := {jlu~ > 0}
the inequality d~Vx~L(x, y °, u)d > 0
is fulfilled.
Moreover, if (A1) and (A3) are satisfied, then the function x(.) is continuous at y0 and directionally differentiable [3], i. e.
x'(y°; r) := lim t-l[z(y ° + tr) - x °] t--~o-t-
e x i s t s for each direction r. This directional derivative is equal to the unique optimal solu- tion of the following quadratic programming problem (Q(u, I, r)) for some u E U(x °, yO) and a certain set l,J+(u) C. I c. I(x°,y°):
1 o o ~d~V~L(x , y , u)d + d~V~yL(x °, yO, u)r --* mind
{ = 0 ~ i E I , V~h~(~o, N0)d + V,h~(~0,yo)~ < 0, i e X(x °, N °) \ I.
If assumption (A2) is also satisfied, then for each direction r there exists a vertex u ° = u°(r) E EV(x °, yO) such that x'(y°; r) is the unique optimal solution of (Q(u °, J+(u°), r)). It has also been shown in [3] that the directional derivative x'(y°; r) is in general not
20
equal to the optimal solution of (Q(u, J+0'),')) for each verte× ~ e EU(: °, yo). So, the following question arises: What is the nature of the elements of the set
f~(,.) = { dl 3u ° E EU(z°,v °) such that d equals } the optimal solution of (Q(u °, J+(u°), r)) "
The following thcorcm shows the relation of these elements to the contingent derivative of the function x(-) at y0 in direction r which is defined as
Duox(r) = { wl 3{(wk, rk, tk)}~'=, converging to (w,r,O+) and } satisfying x(yO + t~.rk) = xo +t~ .wk , k = l , 2 , . . . •
Theorem 2.1 ([3]) Consider problem (I) at y = yO and let x ° E ~(yO). Assume (.41)- (AS) to be satisfied. Then, for each fixed direction r
(i) problem (Q(u, g+(u), r)) has a unique optimal solution iff
u e Argmax{VvL(x°,y°,v)rlv E U(x°,y°)}.
(ii) Dyox(r) C_ f~(r).
(iii) I f also the assumption
(A4) The gradients {Vhi(x°,y°)[i e I(x°,y°)} are linearly independent
is satisfied, then Dyox(r) = l~(r).
Remark 2.2 Under the assumptions of Theorem 2.1(ii), the sets D~x(r) and I2(r) are both finite . Both may differ only for solutions d of problems (Q(u, d+(u), r)) for which there is no sequence {(u k, r k, tk)}~=l converging to (u, r, 0+) with u k E U(x(y°+tk'rk), yo+ t k. rk), k = I, 2, . . . . This is in general not the case if assumption (A2) is dropped.
3 A n e c e s s a r y o p t i m a l i t y c o n d i t i o n
Now, consider problem (1),(2). Let the functions f0 , f be smooth. Then, the objective function g of problem (3) possesses a contingent derivative Dy0g(r) which contains only finitely many elements provided that function x(.) has this property. The following theo- rem gives a necessary optimality condition for problem (3) which will be used to describe a necessary optimality condition for problem (1), (2) in what follows.
Theorem 3.1 Let yO be a local optimal solution of problem (3). Then, there is no direc- tion r such that the following system has a solution:
~o < o, (5)
VufdCy°)r < O, i e I°(y °) :-- {jlfjCy °) = 0}, (6) ~ oyogCr). (7)
21
Proo f : Let there exist a direction r ° such that system (5)-(7) has a solution. Then, there is a 6 > 0 such that
Vj i (y° ) r ° <_ -6, i E I°(y°). (8)
Moreover, since finiteness of Dyo(r °) and by (5),(7), we may assume that there exists w ° ~ Dvo(r °) satisfying
w ° _< -6 . (9)
Now, by definition of D~0 (r°), there must be a sequence {(w k, r k, tk)}~°=l converging to (w °, r °, 0+) with
g(yO + t~. r~) = g(y0) + t k" wk < g(yO)_ tk. 6/2 < g(yO)
for sufficiently large k. Due to (8) , the point yk = yO + tk. rk is also feasible for (3). Thus, yO cannot be an optimal solution of (3). o Now, our aim is it to apply Theorem 3.1 to problem (1),(2). Clearly, if assumptions (A1)-(A4) are satisfied, then, by Theorem 2.1(iii),
Duog(r) = {V~fo(x °, y°)d + Vvfo(x °, y°)rld E f/(r)}. (10)
Thus, by usual methods and using the definition of f/(r), the existence of ~ solution to system (5)-(7) can be verified by solving a certain bilevel progamming problem. If problems" (Q(u,J+(u),r)) are replaced by their necessary and sufficient Kuhn-Tucker conditions and if an active-set strategy is also used, the optimality condition of Theorem 3.1 will be trausformcd into the following condition:
Co ro l l a ry 3.2 Let yO be a local optimal solution of problem (1),(2). Furthermore, let x ° E kO(yO) and assume (,41)-(A4) to be satisfied at (x °, yO). Then, the following combinatorial optimization problem will have a nonnegative optimal value v~:
v,~ := min{v~(u,I)lu E EU(x°,y°),J+(u) C_ I C_ I(x°,y°)}, (11)
where v~,(u, I) denotes the optimal value of the problem
a ~ min (12)
V~fo(x °,y°)d + V j o ( X °,y°)r < a, (13) v~f,(y°)~ < ~, i ~ ~o(yo), (14)
V~L(x°,y°,u)d + XTxyL(x°,y°,u)r + E viV~hi(x°,y°) = 0, (15) iEI
= 0, i e I, (16) V~h,(xo, yO)d + V~h,(x0, y0)~ < 0, i e I(~ °, yo) \ I,
v, _> o, i e x \ J+(~,), Ildl < l. (17)
22
This optimality condition is illustrated by the following example, the lower level problem of which is borrowed from [10].
E x a m p l e 3.3 Consider problem
min{--2zx + 2z2 + 2yl - y~J - y, < 0, z q ~(y)},
where • 1 1 2
k0(y) = Argm~n{~(xx - 1) 2 + ~z21xx < 0,zt + z2yt + Y2 < 0}
at yO = (0, 0) ~. Then,
z ° = (0, 0)', EU(z °, yO) = {(I, 0)' , (0, i)"}, I(z °, yO) = {I, 2}.
Here, for u = (1,0) ' , I = {1}, problem (12)-(17) is given by
min{a[ -2da + 2d2 + 2rl - r2 _< a , - r x _< c~, d~ + v, = 0 ,dz = 0 , d , = 0,dx + r~ ___ 0, Ilrll --- 1}.
This problem is equivalent to
min{al2r, - rz _< a , - r x < a, r2 ~ O, Ilrll ___ a}.
The optimal solution of this problem is
(4 r, a, ,,) = ((0, 0) ' , (0, 0)' , 0, (0, 0)%
Analogously, for u = (0, 1)', I = {2}, we obtain the problem
min{alO < r2 < a , - r x < a, Ilrtl -< 1}
having e.g. the optimal solution
(d,r,~,v) = ( 0 , - - 1 ) ' , ( 1 , 0 ) ' , 0 , ( 0 , 0 ) r ) .
Thus, the necessary optimality condition is satisfied• But, setting y = (Ya, Y~)', we have z(y) = (0 , -Vl)" and g(y) = -y~ < 0 for Yl > O. Consequently, yO = (0,0)" is not locally optimal.
R e m a r k 3.4 If the directional derivative x'(y°; .) is continuous with respect to perturba- tions of the direction, inequality (14) is allowed to be replaced by
v~,h(v°)r < o, i E P(v°). (18)
Let ~o(u, I) be the optimal value of the problem (12), (13), (18), (15)-(17) and let
% := min{%(u,I)lu E EU(z°,y°),J+(u) C_ I C_/(z°,U°)}.
Then, % <_ v~,. Moreover, if z'(y°; .) is continuous, % >_ 0 is a necessary optimality condition. But, in 9eneral, rio may be less than zero, even if yO is a locally optimal solution of problem (1), (2).
23
4 A sufficient optimality condition
Theorem 4.1 Consider problem (I), (P) at y = yo and let x ° e k~(y°). Assume (AI)- (A$) to be satisfied at (x °, yo) and denote the optimal value of problem (1P), (13), (18), (15), (1~),
v, > O, i E I \ J+(u), VII = 1 (19)
by vo(u, I). Then, if
vo := min{vo(u, l)lu E EU(x °, yO), J+(u) C I C I(x °, y0)} > 0, (20)
then, for each 0 < t~ < Vo there exists an open neighbourhood B(y °) of yO such that
fo(x(y), y) > fo(x °, y °) + filly - v°ll
yo,- each V satisfying f (v) <- O, V e B(V°).
By
lim inf t -l[f0(x(y ° + tr), yo + tr) - fo(x °, yo)] t - - , 0 + , r . - , f ~
= min{V=fo(x°,v°)dld e a(e)} + V~fo(z°,y°)f,
the assertion of the theorem is a straightforward consequence of the sufficient optimality conditions derived in [11]. The above example shows that condition (20) may not be substituted by vo > 0, in the general case. This substitution is also not allowed if z'(y °, .) is continuous everywhere. It is not very difficult to construct examples showing vo > 0 at a nonoptimal solution. An exception, where v0 > 0 is indeed a necessary and sufficient optimality condition, are problems in which z(.) is piecewise affin-linear.
References
[1] 3.-P. Aubin and I. EkelaJnd, Applied Nonlinear Analysis, Wiley-Interscience, New York, 1984.
[2] S. Dempe, On the directional derivative of the optimal solution mapping without linear independence constraint qualification, Optimization 20(1989)4, 401-414 (with a Corrigendum in Optimization 22(1991)3, 417)
[3] S. Dempe, Directional differentiability of optimal solutions under Slater's condition, accepted for publication in Math. Programming.
[4] J. Gauvin, A necessary and su~cient regularity condition to have bounded multipliers in nonconvez programming, Math. Programming 12(1977)1, 136-139.
24
[5] R. 3anin, Directional derivative of the marginal function in nonlinear programming, Math. Programming Stud. vol. 21 , 1984, 110-126.
[6] M. Kojima, Strongly stable stationary solutions in nonlinear prograrns, in: Analysis and Computation of Fixed Points, S.M. Robinson, ed., Academic Press, New York, 1980, 93-138.
[7] R. Lucchetti, F. Mignanego and G. Pieri, Existence theorem of equilibrium points in Stackelberg games with constraints, Optimization 18(1987)6, 857-866.
[8] J.P. Penot, A new constraint qualification, J. Optim. Theor. Appl. 48(1986)3, 459- 468.
[9] S.M. Robinson, Generalized equations and their solutions. Part II: Applications to nonlinear programming. Math. Programming Stud. vol. 19, 1982, 200--221.
[10] A. Shapiro, Sensitivity analysis of nonlinear programs and differentiability properties of metric projections, SIAM J. Control Optim. 26(1988)3, 628-645.
[11] M. Studniarski, Necessary and sufficient conditions for isolated local minimum of nonsmooth functions SIAM I. Control Optim. 24(1986)5, 1044-1049.
[12] D.E. Ward, Directional derivative calculus and optimality conditions in nonsmooth mathematical programming. 3. Inform. Optim. Sei. 10(I989)1, 81-96.
A T~SFORMATION ~DR SOLVING A DISCRETE-TIME
SINGULAR LQ PROBLEM x)
Silesian Technical University, Institute of Automatic Control, ul. Pstrowskiego 16, ~4-I01Gliwice, Poland
Summa~. A Linear-~adratlc (LG) dlscrete-tlme problem with singua~eighting matrix of the controls in the performance index is considered . The transformation of the state is proposed for sol- ving the considered problem . The transformation gives the converted state equations having,partlally the Luenberger-Brunovsk# controllab- le canonical form. Using this form the transformed nonsingular LQ problem with inconstant dimensions of state and control is constructed.
Key-words. Optimal control ; discrete-time systems ; canonical forms ; singular problems .
I. Introduction
In engineering, the cases in which to some components of the con-
trol are related no costs are rather frequent . In such cases the
corresponding to them the LQ problems may be singular . The singular
control problems for continuous time systems were considered in many
papers and books e.g. [2,3] •
At the same time there exist not many papers related to singular
discrete-tlme LG problems. However in [2] the latter problems are
discussed and the so called constant directions of the Riccati equa-
tion are researched and exploited in order to reduce the dynamic
order of this equation. In these considerations the problem of
singularity plays not a crucial role .
In the present paper, it has been stressed that the singularity
of the discrete-tlme LQ problem causes some calculatlonal difficul-
ties. The proposed here transformation of the state, similar to that
of continuous time systems [3] converts in one step the singular
problem to nonsingular one.
X) The paper was supported by the departmental program No RP.I.02, coordinated by the Institute of Automatic Control of the Warsaw Technical University .
26
Let us consider the problem described by the following difference
equation and performance index
[xTCtlQxCt) + 2xTCtlGuCt) + uT(tlP)/(t)] (21 t=O
x and u are n and r-dimensional vectors of state and
control, respectively ; A,B,Q,G,H are appropriate constant matrices [o :] and the matrix B is of full rank ; the matrix , as well as
G T ,
H is symmetric, nonnegative definite, and rank H=r-I , where
O<l~<r , l<n ; t--O,1, .., N is the discrete time and N is the
stopping time . The initial state x(0)=x o is given and the final
state x(N+1) is free .
The solved problem is as follows : Among admissible control laws,
the Optimal Control Law (OCL) is to be found for which the perfor-
mance index (2) takes the minimal value .
We would llke to stress here, that for instance for the matrices
Q such that QB=O the considered problem is singular if the matrix
H has not full rank . This results from the fact that in the corres-
ponding to (1),(2) Riccati equation, for t=N-1 the inverse matrix
H -1 appears and we can not start with the calculations .
3. A L i n e a r Transformation of State
Let 0 u and 0 x be the subspace of
such that
{u : o} {x : o}
Let u=Pu be the linear transformation with the nonsingular r x r
matrix P, such that pTHp - H, where in the last 1 rows and last 1
columns of H , zeros appear. Let Pi' i=1,2, .., r be the i-th
column of the matrix P. Let ~T=[vT , e~, vT= Jut,u2, .., Ud] '
eT [~d+l' ~d÷2' "" at] ' d-r-1 . n~m the as~ptions conce=~ng
H it results that the zero-costs are assigned to the vector e .
Let us take into account the 1 following sequences of the
vectors
27
m.-1 BPi, ABPi, A2BPi , .., A i BPi, i=d+l, d+2, .., d+l=r (4)
where m i is determined by the two following relations
and
mi-1 A BPi ~ 0
m=0,1, .., mi-2 (5)
(6)
as well as mien . Among the vectors (4) of these i sequences we
choose the maximal number of linearly independent vectors in accor-
dance with the following scheme . We start with the vectors BPi ,
i=d+S, d+2, .., r , and then ABPl , i-d+1, d+2, .., r , and then
A2BPi , i=d+1, d+2, .., r , and so forth, until the maximal number,
say h of linearly independent vectors is chosen . By this manner,
from each of the i-th sequence (4) we choose, say n i vectors
(0<ni~m i ) so that all the chosen vectors are linearly independent
and h = nd+1+ nd+2+ ..+ n r. We write the chosen vectors in the
following order
n nr-1 BPd+ 1, ABPd+ I, .., Ad+I"IBPd+I , .., BP r, ABPr, .., A Bp r (7)
and denote appropriately by
Wg+l' Wg+2' "., Wg+nd+ I, "', Wn_nr+l, Wn-nr+2, .., w n
where g=n-h .
independent and independent of the vectors (8). Thus, the matrix
W = [Wl,W2, .., Wn] is nonslngular. The formula x=W~ in which .
(8)
4. The Converted Equations
Applying to (I) and (2) the transformations of state and control
defined by the matrices W and P we obtain
x(t+1) = W-IAWx(t) + '#"IBP~. (9)
N I = ~-- [{T(t)wTQwx(t) + 2{T(t)wTGI~(t) + ~T(t)pTHp~(t) I
t=O (~o)
28
Let x = [ z T, yT]T , z T = [ ~ l , ~ 2 , . . , ~g], yT=[~g+l,~g+2 ' . . , ~n] '
dim y=h , g+h=n . Let us notice that the quadratic fonn ~TwTQwx
of the vector ~ = [z T, yT]T has zero-coefficients related to these
components of y for which the corresponding vector-column w i (8)
of the matrix W fulfils the relation (5). This property results
from the determination (3) of the subspace 0 x . Let s be the
q-dimensional vector (0~q ~l ) composed of all these components of
y which correspond to the vector-columns wj (8) fulfilling the
relation (6) . Therefore, we have
~TwTQw~ + 2~TwTGp~ + ~TpTHp~ = zTQggZ + 2zTQgqS + sTQqqS +
+ 2zTGgd v + 2sTGqd v + vTHdd v (11)
where the rlght-hand side of (11) results from deleting in the matrices wTQw, wTGp, pTHp ,the zero-rows and zero-columns. For example the
bilinear form 2~TwTGp~ has zero coefficients corresponding to the
vector e . Really, in the opposite case the quadratic form (11) would
take negative values for some e since the quadratic form of e dis-
appears in (11). The indices of the introduced matrices determine
the dimensions of these matrices, e.g. the matrix Qgq has the dimen-
sion g x q . The following theorem can be proved true (see Appendix).
Theorem 1.
i=
~_._x "4 o,. . . ,o,x l VoY.;.,o,x X ...X I I,...,O,XI..10,...,O,X
... I ... I I ...
X ...X I O,...,1,X! ~0,...,O,X ------~ I . "
"-. ' o,.. o,...,o,x I I ...,0,X X .X I 0,. .,0,X I,
i I • "" I "- ! I "'°
g nd+1 nr
applied to the equation (1) give ~=W'IAw and ~--W'IBp i n the form
x...xloJ I,o] } ... I..I . I..I
X'll' ' '°1 t x . . . x , o Io / oq , , , J ,, I
• I " " "" nd+l X ...X~O ! i o
... i..I .... _ - T }
" ' " I I . . I - '
d 1
(12)
29
where in the places of "X" some nonzero elements can appear. The last h columns of ~ similarly as the vectors (8) are divided into 1
groups each of which contains appropriately n i columns,i--d+l,d+2,..
..,r . For these groups of columns for which ni< m i also in the
places of "X" in the first g rows zeros appear.
Remark 1. The matrices A and B in the last h and i columns,
respectively, are similar to those of the Luenberger-Brunovsk# [4,1]
controllable canonical form .
5. Construction of the Transformed Lq Problem
The components of the vector y, similarly as the vectors (8) we
can divide into 1 groups numbered by the indices i= d+l,d+2,.., r.
The i-th group contains n i components; Let (11), (i2), .., (iq)
be all the indices appearing among d+1,d+2, .., r for which
n(ij)= m(ij) , i.e. for i=(iJ) the relation (6) is fulfilled. Let
/ii/,/i2/, .., /iq'/, q'= 1-q be remaining indices for which
n/ij/<m/ij/ . Let us denote
Pd = [PI'P2' "" Pd] 03)
HF~ =
(~)
p•GTWq , P~HP d
as the matrices r x d , n x q and ~ x ~ (~=d+q)-dlmensional,
respectively. Let k= Max n(ij) , J=1,2, .., q .
Let n = n(i 1)+ n(i2)+ .. + n(lq) , n~= n/il/+ n/12/+ ..+ n/iq./,
where n+n~=h . Let y and y~ be n and n~-dimenslonal vectors
composed of these components of y which belong to the groups numbered
by the indices (11),(i2), .., (iq), and /11/,/12/, .., /lq°/-
-respectively .
A I : q=l . n
A 2 : Each vector A /iJ/BP/ij / , J=1,2, .., q° is linearly dependent
30
on the ( n/j1/ + n/n2/ + ..+ n/iqO/)-following vectors :
Am~llJl I Jl " , m=O,1, .., n i -1 , J=1,2, .., q .
k 4 n / i j / + l , J=1,2, .., q'.
The matrix H~ determined by (15) is nonsingular .
In the further considerations, if nothing different is spoken, we
assume that either A 1 or ~ is fulfilled .
If h <n then taking into account (9), (12) we have
zCt+1)= AggzCt) + AgqsCt) + BgdV(t) (16)
where the matrix Agg contains the elements appearing in the first
g rows and first g columns of A , the matrix Bg d - the elements
of the first g rows and first d columns of ~ , while the matrix Agq
is composed of the q columns which can have nonzero elements in
the first g rows and last h columns of A. The performance index (10)
with accounting (11) takes the form
N I = ~-- [zT(t)QggZ(t) + 2zT(t)QgqS(t) + sT(t)QqqS(t) +
t=O
+ 2zT(t)egdV(t) + 2sT(t)eqdv(t) + vT(t)HddV(t)] (17)
In the formulas (16),(17) the vector e disappears, and only these
components of the vector y can appear which create the vector s .
From (9) and (12) it is seen that each component sj(t) of the vector
s(t) can be freely varied by means of the zero-cost control
e(ij)(t-n(ij) ) , j=1,2, .., q . Taking into account (16) and (17), as
well as (9) and (12) the transformed LQ problem can be written in
the form
N I = ~'- [xT(t)O(t)x(t) + 2xT(t)G(t)u(t) + uT(t)H(t)u(t)]
t=0 . . . . . . . . .
where for t~k the dimensions of the vectors and matrices are
constant and we have
A__(t) = Agg , B( t ) " [Bgd , Agq] (20)
Q(t) = Qgg , _G(t)= [ % q , %d] ' _H(t) = Hdd , Gq
Gqd , Qqq ]
For O~t~k-1 the dimensions of the vectors ~(t), ~(t) and all
the matrices change with t , respectively . This is caused by chan-
ging the role which play the particular components sj of the vector
s in the formulas (16), (17) . For time O~t~n(ij)-1 the compo-
nents sj plays the role of the state component together with
the remaining components of the (iJ)-th group . For t~n(ij) the
component s~ plays the role of control, and the state equations
with the remaining components of the (iJ)-th group are neglected.
Thus we have
z(O)] x(o) Ly(°)J , u(O) v(O) , dim x(O) g+n=~ , dim u(O) = d
(21) For time t , 0 <t~<k-1 , the state x(t) is created by the vector
z(t), as well as by the components of these (iJ)-th groups for which
t <n(ij) ; the control u(t) is created by the vector v(t) and by
components sj taken from these groups for which t~n(ij) , i.e.
we have
dim x( t )= g + ~ n ( i j ) [1-11 ( t - n ( i j ) ) ] , dim u(t)= d+ j =1 - j =1
~ (t'nij) )
( 2 2 )
where I (t)=O for t <0 and I (t)=1 for t~O .
For times O~t~<k-1 the form of the matrices A(t) and B(t)
with the dimensions dim x(t+1) x dim x(t) and dim x(t+1) x dim u(t)
result from taking into account the appropriate elements of the mat-
rices A, B (12) . The form of the matrices Q_(t), G_(t), H(t) result
from successive changing the role of the components sj .
32
Theorem 2.
Let the assumption A 4 be fulfilled . Then the transformed LQ
problem (18), (19), which for t~k takes the form (16), (17) and
has the reduced (n-h)-dlmenslonal state z and the (d+q) dimensional
control [~] is nonslngular .
Proof. Results directly from the assumption A 4 .
It should be noticed that both the LQ problems, the original (1),
(2) and the transformed (18), (19) are equivalent in the sense,
that the variables appearing in these problems are related by the
introduced transformations o~ the state and control. The problem (18),
(19) can be solved by using the conventional formulas .
To sum up , the considered problem can be solved in accordance
with the following scheme ( valid also in the case when the assumption
A 4 is not fulfilled ) :
1. Determine the set 0 u (3) and related transformation matrix P.
2. Determine the set 0 x (3) , create the sequences (4), choose the
linearly independent (of the previously chosen and mutually) vectors
(7) and adjoin them to the previously chosen ones .
3. Determine the indices (i~), J-1,2, .., q for the lastly chosen
vectors (7) .
4. Check, if the assumption A 4 is fulfilled . If yes, construct the
state transformation x&W~ . If not, repeat the points I-4 with
H replaced by ~(k) determined by (20) .
5. Derive the transformed formulas (9), (10) and construct the non-
singular problem (18), (19) with using the indices (iJ) ,
J=1,2, .., q .
It is important that the assumption A 4 can be checked at the time
of creation of the state transformation, without calculation of the
transformed equations (9), (12). It is seen that if A 4 is not fulfi-
lled then proposed state transformation is created recuently, but is
applied only one time . The transformed equation (9) has then the
matrices A,(B) in which the first g,(d) columns of (12) have the
forms similar to those appearing in the last h,(1) columns of (12).
33
If the assumption A 5 is not fulfilled i.e. for some J,m
n/i~/+1 <n(im) = k then the control ~/iO/ can be used for free
varying the appropriate component of s, starting from the instant
n/ij/+1 <n(ij) = k . This will cause the approprla~e change in the
description of the transformed problem (18), (19) in the transient
interval O~ t ~k-1 .
6. Final Conclusions
The singular weighting matrix H of the controls in the performance
index of the considered LQ discrete-time problem can cause some cal-
culational difficulties. In this case the calculations of the OCL
by using the conventional formulas are impossible . The problem can
be solved by using the transformation of state and control similar
to those applied to the continuous case [3] • However in the discrete-
-time case some additional difficulties occur in the initial transient
interval [0, k-l] . These difficulties can be overcomed by the way of
appropriate, successive change of the role of state and control
variables and construction of the transformed nonslngular problem
with inconstant state and control dimensions . The assumption A 4
ensuring the nonslngularlty of the transformed problem can be checked
before the creation of the transformation matrix W . The scheme of
creation of the transformation matrix W is described for the case
when the assumption A 4 is not fulfilled .
References
[I] Brunovsky P. A classification of linear controllable systems. Kybernetika . D.G. CI. 3, 1970, pp.173-187.
[2] Clements A.E., Anderson B.D.O. Singular Optimal Control; The Linear-Quadratic Problem. Lectures Notes in Control and Information Sciences 5, Springer Verlag 1978 .
[5] Gassing H. A Transformation for Solving a singular Linear- -Quadratic Problem. Sent to Journal of Optimization Theory and Applications .
[~ Luenberger D.G. Canonical forms for linear multivariable systems. IEEE Trans. Automatic Control. Vol. AC-12, 1967, pp.290-293.
34
Appendix Proof of the Theorem I.
Let in (I) with x(O)=O we apply u(t)=P~(t) , u(t)= [vT(t),
~d+1(t), Ud+2(t), .., Ur(t)] T , v(t)~O , ~i(0)=1 , uj(O)=O , J~i ,
i,J= d+l, d+2, .., r , ~i(t)~O , t>O , then we have
x(t)= At-lBPi , t~1 , i=d+l, d+2, .., r (1")
and
x(t) = At'15 i , t>1 , i=d+l, d+2, .., r (2")
where 5 i is the i-th column of ~ . From the other hand we have
xCt)= wxCt) (3")
Taking into account the determination of the matrix W we obtain,
that the dependence (3") can be fulfilled with x(t) and x(t)
determined by (1") and (2") then and only then when the matrices
and B have the form (12). Additionally, if for some i we have
n i <m i then the vector x(ni)= --A*i-IBPi resulting f~om (1") is
linearly dependent on the vectors (7), which results from the method
of choice of (7) • Taking into account also (2") and (3 ° ) we obtain
that in the last "X" column of the group for which ni< m i , in
the first g rows zeros must appear Q.E.D.
F A S T S O L U T I O N OF G E N E R A L
N O N L I N E A R F I X E D P O I N T P R O B L E M S
Roberto L. V. Gonz£1ez, Mabel M. Tidball
Departamenlo de Matematica. Faeultad de Cisncias Exactas, lngenierla y Agrimensnra Universidad National de Rosario, Avda. Pellegrini 250, 2000 Rosario, Argentina
ABSTRACT
In this paper, we develope a general procedure to stabilize the usual Newton method in such a way that algorithms obtained always converge to the unique s~lution of the problem. We show that in the case where the operator T ~ C 1 NH z '°° , quadratic convergence holds and when T is polyhcdric, convergence in a finite number of steps is obtained. Numerical results are shown for an example issued from the field of differential games.
I. INTRODUCTION
Frequently, optimal control problems and differential games problems originate variational inequalities (see
[4] and [7]). Also, problems issued from other fields axe reduced to these type of inequalities. In order to
obtain numerical solutions, it is necessary to discretize the original problem (tile continuous solution is
contained in an infinite dimensional space); in that way, the final problem, which must he solved
eomputationally, is reduced to find the fixed point" of a contractive operator. When the actualization rate of
tbe original problem is small (see [5]), the numerical resolution (found by relaxation type iterative algo-
rithms, see [1]) may lead to procedures with slow convergence. In [5], [6], we have introduced acceleration
procedures to improve the speed of convergence of the usual algorithm of Picard type; it essentially consists
of the combination of Picard's and Newton's methods. In this paper, we extend the procedures presented
there, in order to make them applicable to general nonlinear contractive operators.
The set of results obtained is the following. In a first place we have developed a general procedure to
stabillze the usual Newton method in such a way that algorithms obtained always converge to the unique
solution of tile discrete problem (in particular, this technique enables us to transform Howard's methods,
which are not convergent in the case of general differential games problems [11], and to make them
applicable to others problems outside the original fields of application). In general, although tile modified
Newton's algorithm is convergent, no improvement of the order of speed ofconvergence can be expected; in
36
fact we give an example where independently of the chosen starting point, the convergence is geometric of
order I]3. In spite of these negative results, two fields of successful application are shown: the case where the
operator T E C1~I I 2'°° and the case where T is polyhedric. In the first ease, quadratic convergence in
proved; in the second one convergcnce in a finite number of steps is obtained. Finally, numerical results are
shown for an example issued from the field of differential games.
2. PROBLEM DESCRIPTION
Let T be an operator defined in ~n, such that
T e 0°[7 t l l 'C°(~ n)
We assume that operator T is contractive, i.e. there exist p, 0< p < 1 such that T verifies
I T x - - T i g < ( 1 - p ) l x - - i M ¥ x , ~ E %n.
(i)
(2)
Tile algorithms proposed in this paper, are aimed to compute in a fast way the solution of the following
problem:
P: F i n d T E ~ n, such that T~=W. (3)
As T is contractive, it follows that there exists an unique solution for (3).
2.2 Iteratlvo Computation of the Fixcd Point
The Fixed Point Theorem gives us the following algorithm for the computation of ~:
A0 algorithm:
Step 2: compute xU+l----- Tx v
Step 3: if x v = x v+ l then, stop; else, set u = u + l and go to Step 2.
For the convergence of algorithm A0 tile following result holds (sce [1]):
Theorem 2.1: AO algorit/,m produces either a finite sequence x v whose last clement is the exact solution Y~ of
the problem, or generates an infinite sequence z v converging to ~. Also, tl, e following bound for the
approzimation error is valid:
3. AN ABSTRACT ALGORITHM AND ITS CONVEKGENCE
3.0 Preliminary Discuslon.
Although algorithm A0 converges from any arbitrary initial point x 0 , the corresponding speed of
convergcnce is very slow when factor p tcnds to zero. To accelerate this procedure, Newton's type methods
should be used. But in general, these methods are not convergent from everywhere and in consequence, it is
37
necessary to design a technic to stabilize them and to achieve global convergence, (sce for instance the
appendix in [6], where it is shown that a particular case of Newton's methods, IIoward's method; originally
introduced to solve optimal control problems, may be not convergent when it is applied to solve differential
games problems).
To stabilize the method we use a merit function which measures the distance from the current point
x to the solution g . The special algorithm presented here generates a sequence of points x u such that the
associated sequence V(x v ) is a monotonously decreasing sequence converging to zero. This procedure is
obviously related to Lyapnnov's methodology to stabilize dynamical systems. (see for illustrative remarks
about this fact, the clever introduction of the book of Polak [9])
3.1 Lyapunov's Function. Equivalent Problem.
We define, in a natural way, the following Lyapunov's function
V(x) = 1 T x - x |2
The function V satisfies the following properties:
V ( x ) = 0 ¢~ x = T x
V(x) > p 2 1 x - x 12
where ~ is the solution of the problem.
(5)
(6) (7)
We are now iu conditions to introduce the auxiliary problem P':
(s)
It is obvious, by (6) and (7) that problems P and P' are equivalent in the sense that both of them have the
same solution g .
3.2 Abstract Algorithm.
We define here a general algorithm and we prove the convergence in terms of the descent of Lyapunov's
function.
M: ~n _, p(~n)
We shall suppose that M decreases the value of V in the following sense:
V(y) < 9' V(x) V y ~ Mx (9)
w h e r e 0 < 7 < I.
Algorithm Aa
Step 1: set x°E~3t, n, and v = 0 .
Step 2: choose xV+lE Mx v
Step 3: i fx v =-x v + l , then stop; else set v = v + l and go to Step 2.
The convergence of algorithm Aa is assured by condition (9), as it is established in the following:
3 8
Theorem 3.1: If V(y) < 7 V(x) V y ~ M x with 3'< 1, then the abstract algorithm Aa giees lhe solution R in
a finite number of steps or generates a sequence conuerging to X.
3.3 Necessity of Condition V(M(x)) < 7 V(x).
In algorithm Aa, condition
V(y))_<TV(x ) V y ~ M(x)
cannot be replaced by. V(y ) < V(x) without losing the property of convergence,
In effect, if we consider the following function T : ~ --~
and
2 [ _(3~+___1)~ +x if x> x if - l < x < l T(x)= 0 2
( 1 4 3 x ) ~ + x if x < - 1
(10)
MCx) = 0 if x e [-1,1]
MCx) = x - (1-T'Cx)) ' l (T(x) - x) i fx ~ [-1,1] xv oo we obtain that: if [Xo[ > 1, algorithm Aa generates a sequence { )v=t such that, although sequence I x v [
is decreasing, it is not convergent to zero (in fact, ]xVl = 1 + ([ xOl - 1) 2"v). That sequence has two
cluster points, 1 and --1, while sequence V(x v) converges monotonously to 1.
3.4 Practical Algorithms.
We have presented above the general algorithm Aa that converges from everywhere. Now we shall define two
practical implementation of it, algorithms AI and A2, trying that these algorithms apply, whenever possible
or convenient, Newton's method to solve the non linear equation T x - x = 0.
This situation is detected testing the descent of Lyapunov's function V. When Newton's method does not
produce a decrement of V, Newton's direction and direction " ix - -x (given by algorithm A0) are associated,
until the new computed point x v+l satisfies condition V (x v+t ) < 7V (x v).
In A1 algorithm, this condition is defined in Step 3, and it involves the computation of T(T(x)). Algorithm
A2 avoids the computation of T(T(x)), using an adaptatlve estimation of factor 7.
3.4.1 Preliminaries for the application of Newton's method.
Definition of the set of "differentials" O(x).
As operator T is Lipschitz continuous, it is only almost everywhere differentiable. In order to define
algorithms AI and A2 (introduced in the following section), it is necessary at every point of ~n to define
generalized linear operators (in fact, we use here a restricted version of Clarke's subdifferentlal or
perldifferential of T at x, for details and a discussion about this matters see [2 D. With this aim, we introduce
the following concepts:
O(x) = {T ' (x )} i f T is differentiable in x
()(x) = 0 i f T is not differentiable in x
Definition 2
®(x)= N U {¢(y)/ly-xl<_.~} (ll) ~_0 Y
By (I) and (2) we have that T is differentiable almost everywhere and that in any point where T is
differentiable it is satisfied that
IT'lco _< t -p , in consequence it is easy to prove the following properties:
O (x) m T ' (x ) if T is continuously differentiable at x
O ( x ) :/: O V x
V r e O ( x ) , l r l _ < t - - p (12)
3.4.2 Algorithm A1. Definition and properties.
Algorithm A1
Step 0: Give a sequence Ap , ~Tp / A1 = I, r h = 0, Ap --~ 0, ~Tp "-' 1 as p - - co
set v :