PARETO OPTIMALITY, GAME THEORY AND...

EQUILIBRIAPARETO OPTIMALITY GAME THEORY AND

Managing Editor Panos M Pardalos (University of Florida)

EditormdashCombinatorial Optimization Ding-Zhu Du (University of Texas at Dallas)

Advisory Board J Birge (University of Chicago) CA Floudas (Princeton University) F Giannessi (University of Pisa) HD Sherali (Virginia Polytechnic and State University) T Terlaky (McMaster University) Y Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades New algorithmic and theoretical techniques have been developed the diffusion into other disciplines has proceeded at a rapid pace and our knowledge of all aspects of the field has grown even more profound At the same time one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field Optimization has been a basic tool in all areas of applied mathematics engineering medicine economics and other sciences

The series Optimization and Its Applications publishes undergraduate and graduate textbooks monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems Some of the topics covered include nonlinear optimization (convex and nonconvex) network flow problems stochastic optimization optimal control discrete optimization multi-objective programming description of software packages approximation techniques and heuristic approaches

VOLUME 1 7

Springer Optimization and Its Applications


ALTANNAR CHINCHULUUN University of Florida Gainesville FL

PANOS M PARDALOSUniversity of Florida Gainesville FL

ATHANASIOS MIGDALASTechnical University of Crete Greece

LEONIDAS PITSOULIS

123

Aristotle University of Thessaloniki Greece

Edited By

copy 2008 by 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

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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

9 8 7 6 5 4 3 2 1

springercom

The use in this publication of trade names trademarks service marks and similar terms even if they are

Managing Editor Editor Combinatorial OptimizationPanos M Pardalos Ding-Zhu Du

University of FloridaGainesville FL

Panos M Pardalos


Altannar Chin chuluun

Athanasios Migdalas

Technical University of CreteGreece Greece

Leonidas Pitsoulis

ISBN 978-0-387-77246-2 e-ISBN 978-0-387-77247-9 DOI101007978-0-387-77247-9

Library of Congress Control Number 2007942554

Aristotle University of Thessaloniki

Department of Industrial

University of Florida Gainesville FL 32611

pardaloscaoiseufledu

Department of Computer Science

University of Texas at DallasR

and Engineering

dzduutdallasedu

Mathematics Subject Classification (2000) 49K35 49N15 49N70 49N75 49N90 90B05 90B0690B10 90B20 90B50 90C05 90C10 90C29 90C31 90C46 90C47 90C90 91A05 91A06 91A10 91A12 91A25 91A35 91A40 91A43 91A46 91A65 91A80 91B06 91B26 91B28 91B50 93A13

and Systems Engineering

Richardson TX 75083

NY 10013 USA) except for brief excerpts in connection with reviews or scholarly analysis Use in

Department of Industrialand Systems Engineering


Department of Production Engineering and Management


Printed on acid-free paper

Cover art entitled NATURE ON PLAY WITH GRAPHICS is designed by Elias Tyligadas

Αεί δέ ὡς πρός εὖ βουλομένους τούς ἐναντίους ἔργῳ παρασκευαζόμεθα καίοὐκ ἑξ ἐκείνων ὡς ἁμαρτησομένων ἔχει δει τάς ἐλπίδας ἀλλ΄ ὡς ἡμων

ἀσφαλως προνοουμένων πολύ τε διαφέρειν οὐ δει νομίζειν ἄνθρωπον ἀνθρώπουκράτιστον δέ εἶναι ὅστις ἐν τοις ἀναγκαιοτάτοις παιδεύεται

Αγόρευσις του Λακεδαιμονίου βασιλέως Αρχιδάμου(ΘΟΥΚΙΔΙΔΟΥ ῾Ιστοριαι Βιβλίον Α΄)

We always base our preparations against an opponent on the assumptionthat his plans are sound indeed it is right to rest our hopes not on a belief

in his blunders but on the soundness of our provisions Nor ought we tobelieve that there is much difference between man and man but to think

that superiority lies with him who is reared in the severest school

Speech by the Spartan king Archidamusaccording to Thoukididesrsquo History of the Peloponnesian War

Translation adapted from wwwwikipediaorg

Preface

Humans have always been involved in situations where decisions must be madethat best fit the circumstances We read for instance in Homerrsquos Iliad1 theoldest written European composition (eighth century BC)

So he taunted Deiphobusrsquo mind was torn ndashshould he pull back and call a friend to his sidesome hardy Trojan or take the Argive on aloneAs he thought it out the first way seemed the bestHe went for Aeneas

The decision taken may or may not affect and be affected by other de-cision makers The best decision may depend on one or more objectives ofthe decision maker The decision may concern a static situation or a situ-ation that evolves in time Thus mathematical and algorithmic tools havebeen developed in order to model analyze and resolve such decision-makingprocesses Mathematical programming multiobjective optimization optimalcontrol theory and static and dynamic game theory provide the language andthe tools to achieve such goals The notions of optimality Pareto efficiencyand equilibrium are intimately related in a mathematical sense and tightlyconnected through the notions of KarushndashKuhnndashTucker (KKT) optimalitycomplementarity variational inequalities and fixed points The problem un-derlying the search for an optimal point an efficient point an equilibrium ora fixed point is essentially the same

It is true that we can recognize in ancient texts the roots for the needof such mathematical formalism It is hard to deny that in the words ofthe Lacedaemonian king Archidamus as given by the historian Thoukidides(fifth century BC) we start to recognize seeds of rationality desired by gametheory2

1Translated by Robert Fagles Penguin Classics2Translation adapted from wwwwikipediaorg

VIII Preface

We always base our preparations against an opponent on the assump-tion that his plans are sound indeed it is right to rest our hopes noton a belief in his blunders but on the soundness of our provisionsNor ought we to believe that there is much difference between manand man but to think that superiority lies with him who is reared inthe severest school

Or that the following verses of the Iliad depict a game situation3

If you really want me to fight to the finish herehave all Trojans and Argives take their seatsand pit me against Menelaus dear to Ares ndashright between the lines ndashwersquoll fight it out for Helen and all her wealthAnd the one who proves the better man and winshersquoll take these treasures fairly lead the woman home

However only with the development of optimization control and gametheory has it been possible to fully achieve the analysis of such and other farmore complicated situations The concepts of equilibrium and optimality areof immense practical importance in decision-making problems of policies andstrategies in understanding and predicting what will eventually happen insystems in different application domains ranging from economics and engi-neering to military applications

This book brings together recent developments in all these fields that sup-port decision making as well as recent applications of these results to a widerange of modern problems The book consists of twenty-nine chapters con-tributed by experts around the world who work with optimization controlgame theory and equilibrium programming either at a theoretical level andorat the level of using these tools in practice Each chapter is of expository butalso of scholarly nature Each includes a state-of-the-art overview relative toits dedicated topic as well as key references in the field The chapters can bedivided into six partially overlapping groups

The first five chapters of the book are concerned with minimax theoryfixed-points and noncooperative game theory The chapter by H Tuy presentsa unified framework for studying existence and stability conditions for min-imax of quasiconvex-quasiconcave functions that refines several known re-sults from game theory optimization and nonlinear analysis The chapter byB Ricceri surveys recent advances in minimax theory including multiplicitytheorems for nonlinear equations and well-posedness results for optimizationproblems The chapter by JBG Frenk and G Kassay gives an overviewon the theory of noncooperative games both zero-sum and nonconstant-sumgames Based on the KKM lemma they provide proofs of existence of saddle-point strategies in the former case as well as of Nash equilibrium strategies

3Translated by Robert Fagles Penguin Classics

Preface IX

in the latter case The chapter by F Szidarovszky gives an overview of theexistence and computation of equilibrium in nonlinear n-person games Thechapter by G Isac develops a new method for the study of existence of fixedpoints for nonexpansive mappings defined on unbounded sets

Cooperative game theory is concerned with situations in which decisionmakers agree to cooperate in order to maximize their profits or minimizetheir costs In the chapter by I Curiel cooperative combinatorial games areconsidered Such games model situations in which the decision makers whoagree to cooperate encounter a combinatorial optimization problem in orderto maximize their profits or minimize their costs Eight cooperative combina-torial games are surveyed and analyzed The chapter by X Deng and Q Fanghighlights the linear and integer programming approaches to cooperative com-binatorial games as well as computational complexity issues The chapter byJM Bilbao et al introduces the notions of bicooperative games and bisuper-modular games and describes several solution concepts for them The chapterby Y Marinakis et al surveys more than thirteen cooperative combinatorialgames and provides insight through numerical examples

The next five chapters are concerned with dynamic systems in partic-ular with differential games and time-dependent equilibria The chapter byA Maugeri and C Vitanza provides a review of the variational inequalityapproach to problems in a variety of fields including traffic networks mod-els dynamic equilibrium problems as time-dependent variational inequalitiespresents existence results and applies infinite dimensional Lagrangian dualityto these inequalities The chapter by PM Pardalos et al deals with differ-ential games of multiple agents in a hierarchical structure setting as well asin a cooperative setting Controllability observability and optimality prob-lems are studied Maneuvers are introduced using fiber bundles The chapterby V Ostapenko is devoted to developing convex analysis concepts in thecontext of pursuit-evasion differential games The notion of matrix-convexsets and H-convex sets are introduced and their properties required for thetheory of differential games are studied The chapter by AA Chikrii pro-vides a general approach to solving game problems when the dynamics of theconflict-controlled process is described by a system with fractional derivativesSolutions to such systems are derived and sufficient conditions for terminationin guaranteed time are obtained The chapter by M-G Cojocaru et al es-tablishes the equivalence between the solutions to an evolutionary variationalinequality and the critical points of a projected dynamical system in infinitedimensional spaces A convergent algorithm is derived for the solution of evo-lutionary variational inequalities and it is illustrated for the case of trafficnetworks

Information is crucial in the process of decision making The next twochapters are largely concerned with the role and implications of informationin audit policies and auction design In the chapter by K Chatterjee et al asimple auditing model is constructed and analyzed in order to address threeprinciple issues the information contained in the report the commitment to

X Preface

the audit policy and the audit effort The approach is based on the conceptof perfect Bayesian equilibrium An auction is a game with partial informa-tion where an agentrsquos valuation of an object is hidden from other agents Thechapter by RL Zhan provides a thorough survey on the current auctions de-sign literature and synthesizes the developed theories underlying traditionalauctions with new elements and phenomena from emerging and rapidly grow-ing areas such as online auctions

The next five chapters are concerned with multiobjective optimizationbilevel optimization and linear complementarity problems The chapter byG Zhang et al develops a fuzzy multiobjective linear bilevel model to handlehierarchical situations where uncertainty is present in the parameters of eitherthe objective functions or the constraints of the leader and the follower andwhere the leader and the follower may have multiple objectives They derivetheorems characterizing the solutions and develop an approximation KuhnndashTucker approach to solve the problem The chapter by DT Luc is devoted tothe theory of Pareto optimality and discusses existence optimality in productspaces scalarization via support functions duality and solution methodsMultiobjective optimization is overviewed in the chapter by M PappalardoTheorems of solution existence as well as optimality conditions and solutionmethods are presented In the chapter by R Enkhbat et al the weighted sumapproach to finding Pareto optimal solutions in multiobjective optimization isstudied in the context of one-parametric optimization techniques The chapterby B De Schutter is devoted to the linear complementarity problem and toits most general linear extension A link is established between the extendedproblem and max-plus equations that allows the application of the extendedmodel in the analysis and control of discrete-event systems such as trafficsignals manufacturing systems railway networks etc

The remaining eight chapters are largely devoted to applications Fivechapters are devoted to network applications one to military application andtwo to supply chain management The chapter by M Florian and D W Hearnsurveys user equilibrium formulations of static traffic assignment models basedon Wardroprsquos first principle and presents the main solution algorithms for bothdeterministic and stochastic models M Bjoslashrndal and K Joslashrnsten demon-strate in the subsequent chapter that the famous traffic paradox which es-sentially differentiates between Wardropian user equilibrium formulations andnonequilibrium formulations of congested traffic assignment models also oc-curs in congested electricity networks where flows follow Kirchhoffrsquos juctionrule and loop rule Hence it is demonstrated that grid investments may proveto be detrimental to social surplus The chapter by J Cole Smith and C Limexplores models and algorithms applied to a class of Stackelberg (two-stage)games on networks called network interdiction games Two players are in-volved an operator who wishes to execute some function on an existing net-work and an interdictor who acts first to strategically compromise certainelements of the network Recent applications of stochastic models valid in-equalities and bilinear programming techniques to network interdiction games

Preface XI

are discussed and the problem is extended to a three-stage game where theoperator fortifies the network The chapter by M Min reviews game theo-retical approaches in wireless networks addressing mainly the issues of powercontrol cooperation between terminals security and radio channel access con-trol The chapter by D Lozovanu is dedicated to time-discrete systems on net-works where the dynamics of the system is controlled by several players Nashand Pareto optimality principles are applied existence results are derived anddynamic programming techniques are utilized Efficient polynomial-time algo-rithms are developed in order to find optimal strategies of players in dynamicgames on networks G Isac and A Gosselin address a military application ofviability in terms of differential Lanchester type models Set-valued analysisis utilized to introduce the notion of Lanchester type differential inclusionsand to replace the Lanchester coefficients by intervals in order to overcomethe difficulties associated with these coefficients and to facilitate the appli-cation of such models In the chapter by A Nagurney et al static and dy-namic models of global supply chains are developed as networks with threetiers of decision makers A discrete-time algorithm is proposed that allows forthe discretization of the continuous time trajectories The proposed supernet-work framework formalizes the modeling and analysis of global supply chainsThe final chapter by A Chinchuluun et al reviews classic game theoreticalapproaches to modeling and solving problems in supply chain managementBoth noncooperative and cooperative single-period and multiperiod modelsare discussed

Gainesville Chania Thessaloniki A Chinchuluun PM PardalosApril 2007 A Migdalas L Pitsoulis

Contents

Preface VII

List of Contributors XVII

Part I Game and Game Theory

Minimax Existence and StabilityHoang Tuy 3

Recent Advances in Minimax Theory and ApplicationsBiagio Ricceri 23

On Noncooperative Games Minimax Theoremsand Equilibrium ProblemsJohannes BG Frenk Gabor Kassay 53

Nonlinear GamesFerenc Szidarovszky 95

Scalar Asymptotic Contractivity and Fixed Pointsfor Nonexpansive Mappings on Unbounded SetsGeorge Isac 119

Cooperative Combinatorial GamesImma Curiel 131

Algorithmic Cooperative Game TheoryXiaotie Deng Qizhi Fang 159

A Survey of Bicooperative GamesJesus M Bilbao Julio R Fernandez Nieves Jimenez Jorge J Lopez 187

Cost Allocation in Combinatorial Optimization GamesYannis Marinakis Athanasios Migdalas Panos M Pardalos 217

XIV Contents

Time-Dependent Equilibrium ProblemsAntonino Maugeri Carmela Vitanza 249

Differential Games of Multiple Agents and GeometricStructuresPanos M Pardalos Vitaliy A Yatsenko Altannar ChinchuluunArtyom G Nahapetyan 267

Convexity in Differential GamesValentin Ostapenko 307

Game Dynamic Problems for Systems with FractionalDerivativesArkadii A Chikrii 349

Projected Dynamical Systems Evolutionary VariationalInequalities Applications and a Computational ProcedureMonicandashGabriela Cojocaru Patrizia Daniele Anna Nagurney 387

Strategic Audit Policies Without CommitmentKalyan Chatterjee Sanford Morton Arijit Mukherji 407

Optimality and Efficiency in Auctions Design A SurveyRoger L Zhan 437

Part II Multiobjective KKT Bilevel

Solution Concepts and an Approximation KuhnndashTuckerApproach for Fuzzy Multiobjective Linear BilevelProgrammingGuangquan Zhang Jie Lu Tharam Dillon 457

Pareto OptimalityDinh The Luc 481

Multiobjective Optimization A Brief OverviewMassimo Pappalardo 517

Parametric Multiobjective OptimizationRentsen Enkhbat Jurgen Guddat Altannar Chinchuluun 529

Part III Applications

The Extended Linear Complementarity Problem and ItsApplications in Analysis and Control of Discrete-EventSystemsBart De Schutter 541

Contents XV

Traffic Assignment Equilibrium ModelsMichael Florian Donald W Hearn 571

Investment Paradoxes in Electricity NetworksMette Bjoslashrndal Kurt Joslashrnsten 593

Algorithms for Network Interdiction and Fortification GamesJ Cole Smith Churlzu Lim 609

Game Theoretical Approaches in Wireless NetworksManki Min 645

Multiobjective Control of Time-Discrete Systemsand Dynamic Games on NetworksDmitrii Lozovanu 665

A Military Application of Viability Winning ConesDifferential Inclusions and Lanchester Type Modelsfor CombatGeorge Isac Alain Gosselin 759

Statics and Dynamics of Global Supply Chain NetworksAnna Nagurney Jose Cruz Fuminori Toyasaki 799

Game Theory Models and Their Applications in InventoryManagement and Supply ChainAltannar Chinchuluun Athanasia KarakitsiouAthanasia Mavrommati 833

List of Contributors

Jesus M BilbaoDepartment of AppliedMathematics IIUniversity of SevilleEscuela Superior de IngenierosCamino de los Descubrimientos41092 Sevilla Spainmbilbaouses

Mette BjoslashrndalDepartment of Finance andManagement ScienceNorwegian School of Economics andBusiness AdministrationHelleveien 30 N-5045 BergenNorwaymettebjorndalnhhno

Kalyan ChatterjeeDepartment of EconomicsThe Pennsylvania State UniversityUniversity Park PA 16802 USAkchatterjeepsuedu

Arkadii A ChikriiGlushkov Cybernetics Institute NASUkraine40 Glushkov prsp CSP 03680Kiev 187 Ukrainechikinsygkievua

Altannar ChinchuluunCenter for Applied OptimizationDepartment of Industrial andSystems EngineeringUniversity of FloridaGainesville FL 32611USAaltannarufledu

MonicandashGabriela CojocaruDepartment of Mathematics andStatisticsUniversity of GuelphGuelph OntarioCanadamcojocaruoguelphca

Jose CruzDepartment of Operations andInformation ManagementSchool of BusinessUniversity of ConnecticutStorrs CT 06269-1041USAjcruzbusinessuconnedu

Imma CurielUniversity of the NetherlandsAntillesCuracao Netherlands Antillesromaliinterneedsnet

XVIII List of Contributors

Patrizia DanieleDepartment of Mathematics andComputer SciencesUniversity of CataniaViale A Doria 6 95125 CataniaItalydanieledmiunictit

Xiaotie DengDepartment of Computer ScienceCity University of Hong KongHong Kong PR Chinacsdengcityueduhk

Tharam DillonFaculty of Information TechnologyUniversity of TechnologySydney (UTS)Broadway NSW 2007 Australiatharamitutseduau

Rentsen EnkhbatDepartment of MathematicsSchool of Economics StudiesNational University of MongoliaUlaanbaatar Mongoliarenkhbat46sesedumn

Qizhi FangDepartment of MathematicsOcean University of ChinaQingdao 266071 PR Chinaqfangouceducn

Julio R FernandezDepartment of AppliedMathematics IIUniversity of SevilleEscuela Superior de IngenierosCamino de los Descubrimientos41092 Sevilla Spainjuliouses

Michael FlorianCenter for Researchon TransportationUniversity of MontrealMontreal Canada H3C 3J7mikecrtumontrealca

Johannes BG FrenkEconometric InstituteErasmus UniversityRotterdam The Netherlandsfrenkfeweurnl

Alain GosselinDepartment of MathematicsRoyal Military College of CanadaKingston OntarioCanada K7K 7B4gosselin-armcca

Jurgen GuddatInstitute of MathematicsHumboldt University BerlinBerlin Germanyguddatmathematikhu-berlinde

Donald W HearnDepartment of Industrial andSystems EngineeringUniversity of FloridaGainesville FL 32611 USAhearniseufledu

George IsacDepartment of MathematicsRoyal Military College of CanadaSTN Forces Kingston OntarioCanada K7K 7B4isac-grmcca

Nieves JimenezDepartment of AppliedMathematics IIUniversity of SevilleEscuela Superior de IngenierosCamino de los Descubrimientos41092 Sevilla Spainniejimjimuses

List of Contributors XIX

Kurt JoslashrnstenDepartment of Finance andManagement ScienceNorwegian School of Economics andBusiness AdministrationHelleveien 30 N-5045 BergenNorwaykurtjornstennhhno

Athanasia KarakitsiouDecision Support SystemsLaboratoryDepartment of ProductionEngineering and ManagementTechnical University of Crete73100 Chania Greeceatkarverenikeergasyatucgr

Gabor KassayFaculty of Mathematics andComputer ScienceBabes-Bolyai UniversityCluj Romaniakassaymathubbclujro

Churlzu LimSystems Engineeringand Engineering ManagementUniversity of North Carolinaat Charlotte9201 University City BlvdCharlotte NC 28223 USAclim2unccedu

Jorge J LopezDepartment of AppliedMathematics IIUniversity of SevilleEscuela Superior de IngenierosCamino de los Descubrimientos41092 Sevilla Spainjorlopvazuses

Jie LuFaculty of Information TechnologyUniversity of TechnologySydney (UTS)Broadway NSW 2007 Australiajieluitutseduau

Dinh The LucDepartment of MathematicsUniversity of AvignonAvignon Francedtlucunivavignonfr

Dmitrii LozovanuMoldova State UniversityFaculty of Mathematics andInformaticsInstitute of Mathematics andComputer ScienceMoldovan Academy of SciencesAcademy str 5 Chisinau MDndash2028Moldovalozovanumathmd

Yannis MarinakisDecision Support SystemsLaboratoryDepartment of ProductionEngineering and ManagementTechnical University of Crete73100 Chania Greecemarinakisergasyatucgr

Antonino MaugeriDepartment of Mathematics andComputer SciencesUniversity of Catania Viale ADoria 6 95125 Catania Italymaugeridmiunictit

Athanasia MavrommatiDecision Support SystemsLaboratoryDepartment of ProductionEngineering and ManagementTechnical University of Crete73100 Chania Greeceatmavrverenikeergasyatucgr

XX List of Contributors

Athanasios MigdalasDecision Support SystemsLaboratoryDepartment of ProductionEngineering and ManagementTechnical University of Crete 73100Chania Greecesakisverenikeergasyatucgr

Manki MinDepartment of Electrical Engineeringand Computer ScienceSouth Dakota State UniversityBrookings SD USAmankiminsdstateedu

Sanford MortonPrivate ConsultantSeattle WA USAsmortonpoboxcom

Anna NagurneyDepartment of Finance andOperations ManagementIsenberg School of ManagementUniversity of MassachusettsAmherst MA 01003 USAnagurneygbfinumassedu

Artyom G NahapetyanCenter for Applied OptimizationDepartment of Industrial andSystems EngineeringUniversity of FloridaGainesville FL 32611 USAartyomufledu

Valentin OstapenkoTraining-Scientific Complex ldquoInsti-tute of Applied SystemAnalysisrdquoNational Technical University ofUkraine ldquoKyiv PolytechnicalInstituterdquoKiev Ukrainenmommsantu-kpikievua

Massimo PappalardoDepartment of Applied MathematicsUniversity of Pisavia Buonarroti 1c 56127 - PisaItalypappalardodmaunipiit

Panos M PardalosCenter for Applied OptimizationDepartment of Industrial andSystems EngineeringUniversity of FloridaGainesville FL 32611 USApardalosufledu

Biagio RicceriDepartment of MathematicsUniversity of CataniaViale A Doria 6 95125 CataniaItalyricceridmiunictit

Bart De SchutterDelft Center for Systems and ControlDelft University of TechnologyMekelweg 2 2628CD DelftThe Netherlandsbdeschutterdcsctudelftnl

Ferenc SzidarovszkySIE DepartmentThe University of ArizonaTucson AZ USAszidarsiearizonaedu

J Cole SmithDepartment of Industrial andSystems EngineeringUniversity of FloridaGainesville FL 32611 USAcoleiseufledu

Fuminori ToyasakiFaculty of ManagementMcGill UniversityMontreal Quebec H3A 1G5 Canadafuminoritoyasakimcgillca

List of Contributors XXI

Hoang TuyInstitute of Mathematics10307 Hanoi Vietnamhtuymathacvn

Carmela VitanzaDepartment of MathematicsUniversity of MessinaContrada Papardo Salita Sperone31 98100 Messina Italyvitanzacunimeit

Vitaliy A YatsenkoCenter for Applied OptimizationDepartment of Industrial andSystems EngineeringUniversity of Florida

Gainesville FL 32611 USAyatsenkoufledu

Roger L ZhanDepartment of Industrial andSystems EngineeringUniversity of FloridaGainesville FL 32611 USAzhanufledu

Guangquan ZhangFaculty of Information TechnologyUniversity of TechnologySydney (UTS)Broadway NSW 2007 Australiazhanggitutseduau

Part I

Game and Game Theory

Minimax Existence and Stability

Hoang Tuy

Institute of Mathematics Hanoi Vietnamhtuymathacvn

Abstract A unified framework is presented for studying existence and stability con-ditions for minimax of quasiconvex quasiconcave functions These theorems includeas special cases refinements of several known results from game theory optimizationand nonlinear analysis In particular existence conditions are developed that turnout to be sufficient also for the continuity of the saddle value and stability of thesaddle point under continuous perturbation Also a lopsided minimax theorem is es-tablished that yields as immediate corollaries both von Neumannrsquos classic minimaxtheorem and Nashrsquos theorem on noncooperative equilibrium

Key words minimax theorems quasiconvex quasiconcave functions saddlevalue existence conditions stability conditions lopsided minimax coopera-tive equilibrium

1 Introduction

Let XY be two finite-dimensional Euclidean spaces Given two closed convexsets C sub X D sub Y and a function F (x y) C timesD rarr R we define

γ = infxisinC

supyisinD

F (x y) η = supyisinD

infxisinC

F (x y) (1)

We are interested in conditions under which γ = η isin R ie

infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)

If this occurs the common value of γ and η is called a saddle value of thefunction F (x y)

Investigations on the existence of a saddle value for various classes of func-tions were at the beginning motivated by the theory of games According toa classic result of von Neumann [11] later improved by Kneser [9] a saddle

4 H Tuy

value exists when CD are compact convex subsets of X = Rn and Y = R

mrespectively while the function F (x y) is continuous convex in x and con-tinuous concave in y Subsequently it was realized that minimax theoremsalso constitute a very useful tool in different areas of nonlinear analysis andoptimization This has stimulated much research activity over the years forgeneralizing and refining this result

In the first period much effort was spent on relaxing the assumptions onconvexity-concavity and continuity property of F (x y) and also compactnessof both CD The best known result in this direction was Sionrsquos theorem [17]which only required the function F (x y) to be quasiconvex lsc (lower semi-continuous) in x quasiconcave usc (upper semi-continuous) in y and oneof the sets CD (but not necessarily both) to be compact The second periodbegan with the work of Wu [25] who established the first minimax theorem intopological spaces replacing the convexity-concavity assumption by a moregeneral topological property Wursquos theorem required however rather restric-tive assumptions and did not include several minimax results well-known atthe time

In 1974 by a different approach Tuy [19 20] (see also [15 21 24]) proveda topological minimax theorem in the same vein as but much stronger thanWursquos theorem as it did contain most important results currently availablein the field ([9 12 13]) The proof of this theorem besides was very simplemaking use only of elementary set-theoretical arguments

However Tuyrsquos theorem still required compactness of at least one of thesets CD This assumption turned out to be too restrictive for recent de-velopments of mathematical programming and nonlinear analysis (see eg[12] also [614]) To cover the cases considered in these works weaker condi-tions than compactness of C or D had to be developed and quasiconvexity-concavity of F (x y) seemed to be a convenient condition for ensuring existenceof a saddle value when working in vector topological spaces Furthermore con-ceptually all the minimax results so far available for quasiconvex quasiconcavefunctions look somewhat disparate so in [22] an effort has been made to clar-ify the relationship between different existence conditions formulated in thesetheorems and on this basis strengthen and refine several known results

Aside from existence another important topic is stability condition for thesaddle point and continuity property of the saddle value The central resulton this question Golshteinrsquos theorem [5] (see also [16]) though proved morethan three decades ago still remains to our knowledge an isolated result inthis area Although the proof of this theorem is elaborate its assumptions aretoo restrictive if one only needs existence and some weak continuity of thesaddle value rather than these properties for the saddle point

The purpose of the current paper is to provide a sufficiently simple unifiedframework for studying existence and stability conditions for the saddle valueand saddle point of quasiconvex quasiconcave functions and to establish orto refine various strong minimax theorems known to date for this class offunctions As it turns out most of these existence conditions are also sufficientto ensure stability in a sense or another of the saddle value and saddle point

Minimax Existence and Stability 5

After the Introduction in the second section we discuss fundamentalminimax theorems for quasiconvex quasiconcave functions under weakest con-ditions Starting from a basic lemma established by purely set-theoreticalarguments various existence conditions are developed mostly in more or lessrefined form Section 3 deals with continuity and stability of the saddle valueand saddle point under continuous perturbation Some new results are pre-sented that include the above-mentioned theorem of Golshtein as a corollarywhile providing as a by-product a simple proof for this sophisticated the-orem Finally Section 4 presents a new lopsided minimax theorem that isan extension of an ordinary minimax theorem but can also be used to de-rive in a simple way Nash theorem on cooperative equilibrium in n-persongames [10]

Although for the sake of simplicity we restrict ourselves to finite dimen-sional spaces the reader should be aware that many of the results to bepresented can be easily extended to work in a much more general setting

2 Existence Theorems

In this section we discuss conditions to be imposed on the sets CD and thefunction F (x y) in order to guarantee the existence of a saddle value Notethat according to our definition of a saddle value we require that γ = η isin RBecause always γ ge η this excludes the cases γ = minusinfin or η = +infin whichobviously imply that γ = η = minusinfin or γ = η = +infin respectively

Following [1] we say that for a given y isin D the function x rarr F (x y)is lsc (lower semi-continuous) in every line segment if for every a b isin Cthe univariate function φ(λ) = F ((1 minus λ)a + λb y) is lsc on the segment0 le λ le 1

The following lemma is fundamental for deriving the basic existence the-orem which includes virtually all so far known minimax theorems for quasi-convex quasiconcave functions

Lemma 1 (Fundamental Lemma) Assume that the function F (x y) is qua-siconvex lsc in x in every line segment and quasiconcave usc in y Thenfor every nonempty finite set M sub C and every α lt γ we have

capxisinMy isin D| F (x y) ge α = empty (3)

Proof Proceeding by induction we first prove (3) when |M | = 2 For everyx isin C let

D(x) = y isin D| F (x y) ge αBecause γ gt α clearly supyisinD F (x y) gt α forallx isin C and it follows from theassumptions on F (x y) that every set D(x) x isin C is nonempty and closed

6 H Tuy

Arguing by contradiction assume there are a b isin C such that

D(a) capD(b) = empty (4)

Consider a point xλ = (1 minus λ)a + λb with 0 le λ le 1 If y isin D(xλ) then byquasiconvexity of F (x y) in x we have α le F (xλ y) le maxF (a y) F (b y)hence

D(xλ) sub D(a) cupD(b) (5)

Because D(xλ) is convex while D(a)D(b) are disjoint by (4) D(xλ) cannotsimultaneously meet D(a) and D(b) Consequently for every λ isin [0 1] oneand only one of the following alternatives holds

(a) D(xλ) sub D(a) (b) D(xλ) sub D(b)

Denote by La(Lb respectively) the set of all λ isin [0 1] satisfying (a) (satisfying(b) respectively) Clearly 0 isin La 1 isin Lb La cup Lb = [0 1] and analogouslyto (5)

D(xλ) sub D(xλ1) cupD(xλ2) whenever [λ1 le λ le λ2] (6)

Therefore λ isin La implies [0 λ] sub La and λ isin Lb implies [λ 1] sub Lb Lets = supLa = inf Lb and assume for instance that s isin La (the argument issimilar if s isin Lb) We show that (4) leads to a contradiction

We cannot have s = 1 for this would imply D(b) sub D(a) Therefore0 le s lt 1 Because α lt γ le supyisinD F (xs y) it follows that F (xs y) gt α forsome y isin D Because F (xλ y) is lsc in λ there is ε gt 0 such that F (xs+ε y)gt α and so y isin D(xs+ε) But y isin D(xs) sub D(a) hence D(xs+ε) sub D(a) ies + ε isin La contradicting the definition of s Thus (4) cannot occur and sothe proposition holds when |M | = 2

Assuming now that the proposition holds for |M | = k let us prove itfor |M | = k + 1 Let M = x1 xk xk+1 sub C and Dprime = D(xk+1)From the above for any αprime isin (α γ) and any x isin C we have y isinD| F (xk+1 y) ge αprime F (x y) ge αprime = empty hence y isin Dprime| F (x y) ge αprime = empty ie

forallx isin C existy isin Dprime F (x y) ge αprime

which implies that infxisinC supyisinDprime F (x y) ge αprime gt α By the induction hypoth-esis the proposition holds for k points so by applying it with D replaced byDprime we have

capki=1Dprime(xi) = empty

hence capk+1i=1D(xi) = empty

Lemma 2 If Ei| i isin I is an arbitrary collection of closed convex sets in Xwhose intersection is nonempty and compact then there is a finite set J sub Isuch that capjisinJEj is nonempty and compact


Proof Let Ki be the recession cone of Ei and K the recession cone of E =capiisinIEi As is well-known KKi are closed convex cones and K = capiisinIKiLet AAi be the intersection of KKi resp with the unit sphere S = x isinRn| x = 1 Clearly A = capiisinIAi and Ai i isin I are closed subsets of the

compact set S Therefore if E is bounded ie A = empty there must exist a finiteset J sub I such that capjisinJAj = empty Then the closed set capjisinJEj is nonemptyand bounded hence compact

An immediate consequence of the above lemmas is the following basictheorem

Theorem 1 Assume that the function F (x y) is quasiconvex lsc in x inevery line segment for fixed y quasiconcave usc in y for fixed x and inaddition that

(M) There exist a nonempty finite set M sub C and a real number α le γsuch that the set DM = y isin D| minxisinM F (x y) ge α is nonempty andcompact

ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)

Proof Because DM = cap+infink=1D

Mk with DM

k = y isin D| minxisinM F (x y) geαminus 1k and each DM

k is a closed convex set by Lemma 2 there exists r suchthat DM

r is nonempty and compact Therefore by replacing α with αminus 1r ifnecessary we can assume that α lt γ For a fixed natural h take a γh isin (α γ)and consider the sets

Dh(x) = y isin D| minxprimeisinM

F (xprime y) ge γh F (x y) ge γh x isin C

These are all closed subsets of the compact set DM and by Lemma 1 theyhave the finite intersection property Therefore there exists yh isin DM suchthat infxisinC F (x yh) ge γh Noting that DM is compact while the functiony rarr infxisinC F (x y) is usc it then follows that for γh rarr γ the sequenceyh sub DM has a cluster point y isin DM such that infxisinC F (x y) ge γWe cannot have infxisinC F (x y) = +infin because this would imply F (x y) =+infin forallx isin C Therefore γ le maxyisinDM infxisinC F (x y) lt +infin and becausealways maxyisinD infxisinC F (x y) le γ = infxisinC supyisinD F (x y) the equality (7)follows

Remark 1 Condition (M) obviously holds if D is compact while γ lt +infinbecause for any a isin C and M = a sub C the set DM is nonempty This isessentially a boundedness condition for D and in fact it is present in one formor another in all particular minimax theorems so far known for quasiconvexquasiconcave functions

8 H Tuy

Here are various variants of such existence conditions

(Mprime) There exists a finite set M sub C such that for every α isin R the sety isin D| minxisinM F (x y) ge α is compact

( ˜M) There exists a finite set M sub C such that minxisinM F (x y)rarr minusinfin asy isin D y rarr +infin

(H) There exist α isin R such that the set Dα = y isin D| infxisinC F (x y)ge α is nonempty and compact

(Hprime) For every α isin R the set Dα = y isin D| infxisinC F (x y) ge α iscompact

(˜H) infxisinC F (x y)rarr minusinfin as y isin D y rarr +infin(Hlowast) The set Dlowast = y isin D| infxisinC F (x y) = η is nonempty and com-

pact

Theorem 2 Any one of the above conditions implies (M) while

( ˜M) hArr (Mprime) (8)

(˜H) hArr (Hprime) rArr (H) hArr (Hlowast) (9)

Consequently Theorem 1 still holds with condition (M) replaced by any oneof the above conditions

Proof (Mprime)rArr (M) If (Mprime) holds then because the function minxisinM F (x y)is usc (as lower envelope of a family of usc functions in y) while for everyα isin R the set y isin D| minxisinM F (x y) ge α is compact it is easily seen thatthis function has a maximum on D ie maxyisinD minxisinM F (x y) isin R On theother hand from (Mprime) and the fact infxisinC F (x y) le minxisinM F (x y) it followsthat for any α isin R the set y isin D| infxisinC F (x y) ge α is compact Thenthe usc function infxisinC F (x y) must achieve a maximum on D so thatη = maxyisinD infxisinC F (x y) By taking y isin D such that minxisinM F (x y) =maxyisinD minxisinM F (x y) ge η we have y isin DM = y isin D| minxisinM F (x y) geη so the set DM with η le γ is nonempty and bounded ie condition (M)holds

( ˜M) hArr (Mprime) Immediate( ˜M)rArr (H) If ( ˜M)holds then for anyα isin R the set y isin D| minxisinM F (x y)

ge α is compact hence its closed subset y isin D| infxisinC F (x y) ge α toois compact On the other hand for α lt η the latter set is nonempty by thedefinition of η so (H) holds

(H) rArr (M) If (H) holds ie for some α isin R the set Dα is nonempty andcompact then because Dα = capxisinCy isin D| F (x y) ge α by Lemma 2 thereexists a nonempty finite set M sub C such that DM is nonempty and compactfurthermore Dα = empty rArr α le η le γ so (M) holds

(˜H) hArr (Hprime) Immediate


(Hprime) rArr (H) If (Hprime) holds then for every α isin R the set Dα=yisinD|infxisinC F (x y) ge α is compact and for α le η the set Dα is nonempty fromthe definition of η so (H) holds

(H) hArr (Hlowast) Immediate

For any given x isin C we say that the function y rarr F (x y) is uscin every line segment if for any a b isin D the univariate function ψ(λ) =F (x (1minus λ)a+ λb) is usc on the segment 0 le λ le 1 Using the obvious re-lation infxisinC supyisinD F (x y) = minus supxisinC infyisinD(minusF (x y)) we easily deducefrom Theorems 1 and 2 the following propositions

Theorem 3 Assume that the function F (x y) is quasiconvex lsc in x forfixed y quasiconcave usc in every line segment in y for fixed x and inaddition (N) There exist a nonempty finite set N sub D and a real numberβ such that the set CN = x isin C| maxyisinN F (x y) le β is nonempty andcompact

ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)

Theorem 4 Theorem 3 still holds with condition (N) replaced by any one ofthe conditions listed below

(Nprime) There exists a finite set N sub D such that for every β isin R the setx isin C| maxyisinN F (x y) le β is compact

(˜N) There exists a finite set N sub D such that maxyisinD F (x y) rarr +infin asx isin C x rarr +infin(K) There exist β isin R such that the set Cβ = x isin C| supyisinD F (x y)le β is nonempty and compact(Kprime) For every β isin R the set Cβ = y isin D| infxisinC F (x y) le β iscompact(˜K) supyisinD F (x y)rarr +infin as x isin C x rarr +infin(Klowast) The set Clowast = x isin C| supyisinD F (x y) = η is nonempty and com-pact

Remark 2 Most known minimax theorems for quasiconvex quasiconcave func-tions including Sionrsquos well-known result and some refined versions of it as usedin nonlinear analysis (see eg [1 2]) are special cases of the above proposi-tions

Also note that the earliest proofs for minimax theorems used fixed point orseparation arguments in one form or another The above proof given originallyin [19 20] was the first one using only elementary set-theoretical argumentsfor establishing general topological minimax theorems The results in the men-tioned papers with their proofs have been presented partially or in full insome books (see eg [15 24]) Nevertheless exactly the same results were

10 H Tuy

rediscovered in [4] with only a difference of notation Also the above simpleproof was many years later rediscovered in Joo [7 8] according to Frenk andKassay [3]

In the above propositions F (x y) is always assumed to be lsc in x uscin y We now prove some minimax theorems for functions F (x y) lsc ineach variable or usc in each variable

Lemma 3 Assume that the function F (x y) is quasiconvex usc in x in everyline segment for fixed y and quasiconcave usc in y for fixed x If condition(M) in Theorem 1 is satisfied then for any αprime isin (α γ) the family of sets

D(x) = y isin DM | F (x y) ge αprime x isin C

have the finite intersection property

Proof The proof is similar to that of Lemma 1 with the following change inthe argument for showing that (4) cannot occur

For every y isin D(b) because s isin La ieD(xs) sub D(a) we have y isin D(xs)hence F (xs y) lt αprime and by upper semi-continuity of F (xλ y) in λ there existsan open interval Iy = (s1 s2) containing s (s1 = s1(y) s2 = s2(y)) such thatF (xλ y) lt αprime for all λ isin Iy Then F (xsi

y) lt αprime ie y isin D(xsi) i = 1 2

and using the closedness of the sets D(xsi) i = 1 2 we can find for each

i = 1 2 a neighborhood Wi(y) of y such that F (xsi z) lt αprime forallz isin Wi(y)

Clearly Wy = W1(y) capW2(y) is a neighborhood of y such that F (xsi z) lt αprime

for all z isin Wy ie z isin D(xsi) i = 1 2 and hence z isin D(xλ) for all λ isin Iy

Thus for every y isin D(b) there exist a neighborhood Wy and an interval Iysatisfying

F (xλ z) lt αprime forallλ isin Iyforallz isinWy

BecauseD(b) is a closed subset of the compact setDM it is itself compact andfrom the family Wy y isin D(b) one can extract a finite collection Wy y isinE |E| lt +infin still covering D(b) If λ isin I = capyisinEIy and y isin D(b) theny isin Wyprime for some yprime isin E hence F (xλ y) lt αprime Therefore D(xλ) sub D(a) forall λ isin I ie I sub La contradicting the definition of s

Theorem 5 Assume that the function F (x y) C times D rarr R is quasiconvexusc in x in every line segment for fixed y quasiconcave usc in y for fixed xIf condition (M) in Theorem 1 is satisfied then the equality (7) holds

Proof The proof is similar to that of Theorem 1 but using Lemma 3 insteadof Lemma 1 Specifically for a sequence γk isin (α γ) γk γ consider the sets

Dk(x) = y isin D| minxprimeisinM

F (xprime y) ge γk F (x y) ge γk x isin C

For k fixed they are all closed subsets of the compact set DM and by Lemma 3they have the finite intersection property Therefore these sets have a


nonempty intersection ie there exists yk isin DM such that infxisinC F (x yk) geγk Because DM is compact while the function y rarr infxisinC F (x y) is uscit follows that the sequence yk sub DM has a cluster point y isin DM such thatinfxisinC F (x y) ge γ Consequently maxyisinD infxisinC F (x y) ge γ and becausealways maxyisinD infxisinC F (x y) le γ = infxisinC supyisinD F (x y) the equality (7)follows

Theorem 6 Assume that the function F (x y) C times D rarr R is quasiconvexlsc in x for fixed y quasiconcave lsc in y in every line segment for fixed xIf condition (N) in Theorem 3 is satisfied then the equality (10) holds

Remark 3 Propositions analogous to Theorems 1 and 1b can also be derivedfrom Theorems 5 and 6 for example

If F (x y) is as in Theorem 5 but satisfies either (H) or (Hlowast) (instead of(M)) then (10) holds

As a special case of Theorem 6 let us mention the following result ofGolshtein ([5] Theorem 2) which was established (by the way by a ratherelaborate argument) to provide a tool for the foundation of a general dualitytheory of convex programming

Assume that the function F (x y) CtimesD rarr R is convex continuous in x forfixed y and concave in y for fixed x If the set Clowast = x isin C| supyisinD F (x y) =γ is nonempty and compact then the equality (10) holds

Proof A concave function on a convex set is always lsc in every line segmentwhile the assumption about Clowast is nothing but condition (K) in Theorem 4which in turn implies condition (N) in Theorem 3

A point (x y) isin C timesD is said to be a saddle point of F (x y) on the setC timesD if it satisfies

F (x y) le F (x y) le F (x y) forallx isin C forally isin D (11)

As is well-known (see eg [2] Proposition 12 Chapter VI) F (x y) possessesa saddle point on C timesD if and only if

minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)

and then (x y) is a saddle point if and only if (x y) isin Clowast timesDlowast where

Clowast = x isin C| supyisinD

F (x y) = η Dlowast = y isin D| infxisinC

F (x y) = γ (12)

and γ = η is the saddle value

Combining Theorems 1 and 3 yields

12 H Tuy

Theorem 7 Let F (x y) be a function quasiconvex lsc in x for fixed y qua-siconcave usc in y for fixed x Assume that

(MN) There exist two nonempty finite sets M sub C N sub D along withreal numbers α β such that α le γ η le β and the following sets are nonemptyand compact

CN = x isin C| maxyisinN

F (x y) le β DM = y isin D| minxisinM

F (x y) ge α(13)

Then the function F (x y) possesses a saddle point on C timesD

Proof By Theorem 1 maxyisinD infxisinC F (x y) = γ and by Theorem 3minxisinD supyisinD F (x y) = η hence (11)

Corollary 1 With F (x y) as in Theorem 7 if the sets Clowast and Dlowast are non-empty and compact then F (x y) has a saddle point (x y) on C timesD

Proof Then conditions (Hlowast) and (Klowast) hold and this implies (MN) byTheorems 2 and 4

3 Stability Theorems

We now turn to conditions for the existence and continuity of the saddle valueandor the saddle point of a function depending upon a parameter

Let CDX Y be as previously specified let Ω be a metric space andF (u x y) ΩtimesCtimesD rarr R a function continuous on ΩtimesCtimesD quasiconvexin x for fixed (u y) and quasiconcave in y for fixed (u x) For every u isin Ωdefine

γ(u) = infxisinC

supyisinD

F (u x y) η(u) = supyisinD

infxisinC

F (u x y) (14)

It is convenient to begin with a simple fact that will be often needed inthis section

Lemma 4 Let D be a compact set in Y and g(u y) be an usc functionon Ω times D satisfying maxyisinD g(u

lowast y) lt 0 Then there exists an open ball Uaround ulowast such that

maxyisinD

g(u y) lt 0 forallu isin U

Proof By upper semi-continuity for fixed y isin D there exists an open ball Uyaround ulowast and an open ball Vy around y such that g(u yprime) lt 0 foralluisinUyforallyprimeisinVyBecause D is compact a finite set J sub D exists such that D sub cupyisinJVy SettingU = capyisinJUy yields an open ball U around ulowast such that for every y isin D u isin Uwe have y isin Vyprime for some yprime isin J while u isin Uyprime hence g(u y) lt 0


The following theorems have been established in [23] under slightly weakercontinuity conditions for F (u x y) For a given ulowast isin Ω we set

γlowast = γ(ulowast) ηlowast = η(ulowast)

Theorem 8 Assume condition (M) holds ie there exist a nonempty fi-nite set M sub C and a real number α le γlowast such that the set DM (ulowast) =y isin D| minxisinM F (ulowast x y) ge α is nonempty and compact Then γlowast = ηlowast

and there exist a compact set D0 sub D and an open ball U around ulowast such that

empty = y isin D| minxisinM

F (u x y) ge α sub D0 forallu isin U (15)

and the function η(u) = supyisinD infxisinC F (u x y) is upper semi-continuousat ulowast

Proof First by Theorem 1 ηlowast=γlowastNow define ψ(u y) = minxisinM F (u x y)Clearly DM (ulowast) = y isin D| ψ(ulowast y) ge α = cap+infin

k=1DMk where DM

k (ulowast) = y isinY | ψ(ulowast y) ge αminus1k are closed convex sets Hence by Lemma 2 there existsk0 such that DM

k0(ulowast) is compact and by replacing α with αprime = αminus 1k0 lt α

it can be assumed that α lt γlowast and

maxyisinDM (ulowast)

ψ(ulowast y) gt α

Because ψ(u y) is pointwise minimum of finitely many functions continuousin (u y) and quasiconcave in y it is continuous in (u y) and quasiconcavein y Therefore the function u rarr maxyisinDM (ulowast) ψ(u y) is lsc and becausemaxyisinDM (ulowast) ψ(ulowast y) gt α there is an open ball U around ulowast such that


ψ(u y) gt α forallu isin U (16)

In particular DM (u) = y isin D| ψ(u y) ge α = empty forallu isin U Let us show thatthe ball U can be chosen so that all sets DM (u) u isin U are contained in acompact set D0 sub D

Let ˜D = DM (ulowast) It suffices of course to consider the case when D isnoncompact so that D ˜D = empty For δ gt 0 consider the sets

Dδ = y isin D| δ le d(y ˜D) le 2δ D0 = y isin D| d(y ˜D) le 2δ

where d(y ˜D) = minyprimeisin ˜D y minus yprime is the distance from y to the set ˜D

Because ˜D is nonempty by Lemma 1 and compact by assumption Dδ andD0 are also nonempty and compact and we have

ψ(ulowast y) lt α forally isin Dδ (17)

But the function ψ(u y) is usc as it is pointwise minimum of a family ofcontinuous functions so by Lemma 4 there exists an open ball around ulowast

(which can be considered to be the same U) such that

Managing Editor Panos M Pardalos (University of Florida)

EditormdashCombinatorial Optimization Ding-Zhu Du (University of Texas at Dallas)

Advisory Board J Birge (University of Chicago) CA Floudas (Princeton University) F Giannessi (University of Pisa) HD Sherali (Virginia Polytechnic and State University) T Terlaky (McMaster University) Y Ye (Stanford University)

Aims and Scope Optimization has been expanding in all directions at an astonishing rate during the last few decades New algorithmic and theoretical techniques have been developed the diffusion into other disciplines has proceeded at a rapid pace and our knowledge of all aspects of the field has grown even more profound At the same time one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field Optimization has been a basic tool in all areas of applied mathematics engineering medicine economics and other sciences

The series Optimization and Its Applications publishes undergraduate and graduate textbooks monographs and state-of-the-art expository works that focus on algorithms for solving optimization problems and also study applications involving such problems Some of the topics covered include nonlinear optimization (convex and nonconvex) network flow problems stochastic optimization optimal control discrete optimization multi-objective programming description of software packages approximation techniques and heuristic approaches

VOLUME 1 7

Springer Optimization and Its Applications





LEONIDAS PITSOULIS

123


Edited By




9 8 7 6 5 4 3 2 1

springercom




Panos M Pardalos



Athanasios Migdalas


Leonidas Pitsoulis

ISBN 978-0-387-77246-2 e-ISBN 978-0-387-77247-9 DOI101007978-0-387-77247-9








and Engineering

dzduutdallasedu



Richardson TX 75083
















Preface






VIII Preface








Preface IX





X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α
















LEONIDAS PITSOULIS

123


Edited By




9 8 7 6 5 4 3 2 1

springercom




Panos M Pardalos



Athanasios Migdalas


Leonidas Pitsoulis

ISBN 978-0-387-77246-2 e-ISBN 978-0-387-77247-9 DOI101007978-0-387-77247-9








and Engineering

dzduutdallasedu



Richardson TX 75083
















Preface






VIII Preface








Preface IX





X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α















9 8 7 6 5 4 3 2 1

springercom




Panos M Pardalos



Athanasios Migdalas


Leonidas Pitsoulis

ISBN 978-0-387-77246-2 e-ISBN 978-0-387-77247-9 DOI101007978-0-387-77247-9








and Engineering

dzduutdallasedu



Richardson TX 75083
















Preface






VIII Preface








Preface IX





X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α




















Preface






VIII Preface








Preface IX





X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












VIII Preface








Preface IX





X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












Preface IX





X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












X Preface




Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












Preface XI



Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












Contents

Preface VII












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












XIV Contents















Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












Contents XV

































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α




































































Part I



Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α













Hoang Tuy




1 Introduction


γ = infxisinC

supyisinD


infxisinC

F (x y) (1)


infxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) isin R (2)



4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












4 H Tuy



















6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α























6 H Tuy























ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α


















ThenmaxyisinD

infxisinC

F (x y) = infxisinC

supyisinD

F (x y) (7)


Mk with DM








8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












8 H Tuy







pact















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α

















ThenminxisinC

supyisinD

F (x y) = supyisinD

infxisinC

F (x y) (10)






10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












10 H Tuy
































minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α























minxisinC

supyisinD

F (x y) = maxyisinD

infxisinC

F (x y)




F (x y) = γ (12)



12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α












12 H Tuy





F (x y) ge α(13)








γ(u) = infxisinC

supyisinD


infxisinC

F (u x y) (14)




maxyisinD












k=1DMk where DM





ψ(ulowast y) gt α





















k=1DMk where DM





ψ(ulowast y) gt α












PARETO OPTIMALITY, GAME THEORY AND...

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