Optimization with multivalued mappings : theory, applications, and algorithms PDF
Preview Optimization with multivalued mappings : theory, applications, and algorithms
OPTIMIZATION WITH MULTIVALUED MAPPINGS Theory, Applications, and Algorithms Edited by STEPHAN DEMPE TU Bergakademie Freiberg, Germany VYACHESLAV KALASHNIKOV ITESM, Monterrey, Mexico 1 3 Library of Congress Control Number: 2006925111 ISBN-10: 0-387-34220-6 e-ISBN: 0-387-34221-4 ISBN-13: 978-0-387-34220-7 Printed on acid-free paper. AMS Subject Classifications: 90C26, 91A65 (cid:164) 2006 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springer.com This volume is dedicated to our families, our wives Jutta and Nataliya, our children Jana and Raymond, Vitaliy and Mariya Contents Preface ........................................................ ix Part I Bilevel Programming Optimality conditions for bilevel programming problems Stephan Dempe, Vyatcheslav V. Kalashnikov and Nataliya Kalashnykova 3 Path-based formulations of a bilevel toll setting problem Mohamed Didi-Biha, Patrice Marcotte and Gilles Savard.............. 29 Bilevel programming with convex lower level problems Joydeep Dutta and Stephan Dempe ................................. 51 Optimality criteria for bilevel programming problems using the radial subdifferential D. Fangha¨nel.................................................... 73 On approximate mixed Nash equilibria and average marginal functions for two-stage three-players games Lina Mallozzi and Jacqueline Morgan............................... 97 Part II Mathematical Programs with Equilibrium Constraints A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints Michael L. Flegel and Christian Kanzow ............................111 On the use of bilevel programming for solving a structural optimization problem with discrete variables Joaquim J. Ju´dice, Ana M. Faustino, Isabel M. Ribeiro and A. Serra Neves ..........................................................123 viii Contents On the control of an evolutionary equilibrium in micromagnetics Michal Koˇcvara, Martin Kruˇz´ık, Jiˇr´ı V. Outrata .....................143 Complementarity constraints as nonlinear equations: Theory and numerical experience Sven Leyffer.....................................................169 A semi-infinite approach to design centering Oliver Stein .....................................................209 Part III Set-Valued Optimization Contraction mapping fixed point algorithms for solving multivalued mixed variational inequalities Pham Ngoc Anh and Le Dung Muu.................................231 Optimality conditions for a d.c. set-valued problem via the extremal principle N. Gadhi .......................................................251 First and second order optimality conditions in set optimization V. Kalashnikov, B. Jadamba, A.A. Khan............................265 Preface Optimization problems involving multivalued mappings in constraints or as the objective function are investigated intensely in the area of non- differentiable non-convex optimization. Such problems are well-known under the names bilevel programming problems [2, 4, 5], mathematical problems with equilibrium (complementarity) constraints (MPEC) [6, 9], equilibrium problems with equilibrium constraints (EPEC) [8, 10], set-valued optimiza- tion problems [3] and so on. Edited volumes on the field are [1, 7]. Since the publicationofthesevolumestherehasbeenatremendousdevelopmentinthe field which includes the formulation of optimality conditions using different kinds of generalized derivatives for set-valued mappings (as e.g. the coderiva- tive of Mordukhovich), the opening of new applications (as the calibration of water supply systems), or the elaboration of new solution algorithms (as for instance smoothing methods). We are grateful to the contributors of this volumethatthey agreed topublish their newest resultsin thisvolume. These results reflect most of the recent developments in the field. The contributions are classified into three parts. Focus in the first part is on bilevel programming. Different promising possibilities for the construction of optimality condi- tions are the topic of the paper “Optimality conditions for bilevel program- ming problems.” Moreover, the relations between the different problems in- vestigated in this volume are carefully considered. Thecomputationofbesttollstobepayedbytheusersofatransportation network is one important application of bilevel programming. Especially in relation to recent economic developments in traffic systems this problem has attractedlargeinterest.Inthepaper“Path-basedformulationsofabileveltoll settingproblem,”M.Didi-Biha,P.MarcotteandG.Savarddescribedifferent formulations of this problem and develop efficient solution algorithms. Applying Mordukhovich’s coderivative to the normal cone mapping for the feasible set of the lower level problem, J. Dutta and S. Dempe derive necessary optimality conditions in “Bilevel programming with convex lower level problems.” x Preface Most papers in bilevel programming concentrate on continuous problems in both levels of hierarchy. D. Fangha¨nel investigates problems with discrete lowerlevelproblemsinherpaper“Optimalitycriteriaforbilevelprogramming problems using the radial subdifferential.” Important for her developments are the notions of the optimistic resp. the pessimistic optimal solution. Both notions reduce the bilevel problem to the minimization of a discontinuous function. To develop the optimality conditions she uses the analytic tool of a radial-directional derivative and the radial subdifferential. In their paper “On approximate mixed Nash equilibria and average marginal functions for two-stage three-players games,” L. Mallozzi and J. Morgan investigate bilevel optimization problems where the two lower level players react computing a Nash equilibrium depending on the leader’s selec- tion.OtherthaninD.Fangha¨nel’spaper,theyapplyaregularizationapproach to the bilevel problem on order to get a continuous auxiliary function. Thesecondpartcollectspapersinvestigatingmathematicalprogramswith equilibrium constraints. Due to violation of (most of) the standard regularity conditions at all feasible points of mathematical programs with equilibrium constraints using classical approaches only necessary optimality conditions in form of the Fritz John conditions can be obtained. This leads to the formulation of weaker optimalityconditionsasA-,B-,C-andM-stationarity.Thisopensthewayfor theinvestigationofassumptionsguaranteeingthatoneoftheseconditionscan beshowntobenecessaryforoptimality.Intheirpaper“AdirectproofforM- stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints,” M. L. Flegel and C. Kanzow give an interesting proof of such a result. Tosolveandtoinvestigatebilevelprogrammingproblemsormathematical programs with equilibrium constraints these are usually transformed into a (nondifferentiable) standard optimization problem. In “On the use of bilevel programmingforsolvingastructuraloptimizationproblemwithdiscretevari- ables,” J. J. Ju´dice et al. use the opposite approach to solve an applied large- dimensional mixed integer programming problem. They transform this prob- lem into a mathematical program with equilibrium constraints and solve the latter problem using their complementarity active-set algorithm. This then results in a promising algorithm for the original problem. A further applied problem is investigated in the paper “On the control of an evolutionary equilibrium in micromagnetics” by M. Koˇcvara et al. They model the problem as an MPEC in infinite dimensions. The problem after discretization can be solved by applying the implicit programming technique since the solution of the lower level’s evolutionary inequality in uniquely de- termined. The generalized differential calculus of B. Mordukhovich is used to compute the needed subgradients of the composite function. An important issue of the investigations is the formulations of solution algorithms. Promising attempts of applying certain algorithms of nonlinear mathematical programming are quite recent. S. Leyffer investigates in his Preface xi paper “Complementarity constraints as nonlinear equations: Theory and nu- merical experience” one such algorithm. The method is based on an exact smoothing approach to the complementarity conditions. Comprehensive nu- merical tests show the power of the method. Again a challenging applied problem is investigated in the paper “A semi- infiniteapproachtodesigncentering”byO.Stein.Heformulatedthisproblem as a general semi-infinite optimization problem. After certain reformulations, the intrinsic bilevel structure of the problem is detected. The third part on multivalued set-valued optimization starts with the paper “Contraction mapping fixed point algorithms for solving multivalued mixedvariationalinequalities”byP.N.AnhandL.D.Muu.Theyuseafixed- pointapproachtosolvemultivaluedvariationalinequalitiesandBanach’scon- traction mapping principle to find the convergence rate of the algorithm. Mordukhovich’s extremal principle is used in the paper “Optimality con- ditions for a d.c. set-valued problem via the extremal principle” by N. Gadhi to derive optimality conditions for a set-valued optimization problemwith an objective function given as difference of two convex mappings. The necessary optimality conditions are given in form of set inclusions. Second order necessary optimality conditions for set-valued optimization problems are derived in the last paper “First and second order optimality conditions in set optimization” in this volume by V. Kalashnikov et al. The maintheoreticaltoolinthispaperisanepiderivativeforset-valuedmappings. We are very thankful to the referees of the enclosed papers who with their reports for the papers have essentially contributed to the high scientific quality of the enclosed papers. We are also thankful to Sebastian Lohse for carefully reading the text and to numerous colleagues for fruitful and helpful discussions on the topic during the preparation of this volume. Last but not leastwethankJohnMartindaleandRobertSaleyfromthepublisherfortheir continuing support. Freiberg and Monterrey, Stephan Dempe February 2006 Vyatcheslav V. Kalashnikov References 1. G. Anandalingam and T.L. Friesz (eds.) (1992), Hierarchical Optimization, Annals of Operations Research, Vol. 34. 2. J.F. Bard (1998), Practical Bilevel Optimization: Algorithms and Applica- tions, Kluwer Academic Publishers, Dordrecht. 3. G.Y. Chen and J. Jahn (eds.) (1998), Mathematical Methods of Operations Research, Vol. 48. 4. S. Dempe (2002), Foundations of Bilevel Programming, Kluwer Academic Publishers, Dordrecht. xii Preface 5. S. Dempe (2003), Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints, Optimization, 52: 333-359. 6. Z.-Q.LuoandJ.-S.PangandD.Ralph(1996), Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge. 7. A.MigdalasandP.M.PardalosandP.V¨arbrand(eds.)(1998),MultilevelOp- timization: Algorithms and Applications, Kluwer Academic Publishers, Dor- drecht. 8. B.S.Mordukhovich(2003),Equilibriumproblemswithequilibriumconstraints via multiobjective optimization, Technical Report, Wayne State University, Detroit, USA. 9. J. Outrata and M. Koˇcvara and J. Zowe (1998), Nonsmooth Approach to OptimizationProblemswithEquilibriumConstraints,KluwerAcademicPub- lishers, Dordrecht. 10. J.-S.PangandM.Fukushima(2005),Quasi-variationalinequalities,general- izedNashequilibria,andmulti-leader-followergames, Comput. Manag. Sci., 2: 21–56.
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