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Lecture Notes in Economics and Mathematical Systems 583 FoundingEditors: M.Beckmann H.P.Künzi ManagingEditors: Prof.Dr.G.Fandel FachbereichWirtschaftswissenschaften FernuniversitätHagen Feithstr.140/AVZII,58084Hagen,Germany Prof.Dr.W.Trockel InstitutfürMathematischeWirtschaftsforschung(IMW) UniversitätBielefeld Universitätsstr.25,33615Bielefeld,Germany EditorialBoard: A.Basile,A.Drexl,H.Dawid,K.Inderfurth,W.Kürsten,U.Schittko Igor V. Konnov · Dinh The Luc · Alexander M. Rubinov† Generalized Convexity and Related Topics With11Figures 123 ProfessorIgorV.Konnov DepartmentofAppliedMathematics KazanUniversity ul.Kremlevskaya,18 Kazan420008 Russia [email protected] ProfessorDinhTheLuc DepartmentdeMathematiques 33rueLouisPasteur 84000Avignon France [email protected] ProfessorAlexanderM.Rubinov† SITMS UniversityofBallarat UniversityDr.1 3353Victoria Australia [email protected] LibraryofCongressControlNumber:2006934206 ISBN-10 3-540-37006-4 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-37006-2 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationor partsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer-Verlag.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:Camerareadybyauthor Cover:ErichKirchner,Heidelberg Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig SPIN11811275 Printedonacid-freepaper–88/3100YL–543210 Preface In mathematics generalization is one of the main activities of researchers. It opens up new theoretical horizons and broadens the fields of applications. Intensive study of generalized convex objects began about three decades ago whenthetheoryofconvexanalysisnearlyreacheditsperfectstageofdevelop- ment with the pioneering contributions of Fenchel, Moreau, Rockafellar and others. The involvement of a number of scholars in the study of generalized convex functions and generalized monotone operators in recent years is due to the quest for more general techniques that are able to describe and treat models of the real world in which convexity and monotonicity are relaxed. Ideas and methods of generalized convexity are now within reach not only in mathematics, but also in economics, engineering, mechanics, finance and other applied sciences. This volume of referred papers, carefully selected from the contributions delivered at the 8th International Symposium on Generalized Convexity and Monotonicity(Varese,4-8July,2005),offersaglobalpictureofcurrenttrends of research in generalized convexity and generalized monotonicity. It begins withthreeinvitedlecturesbyKonnov,LevinandPardalosonnumericalvaria- tionalanalysis,mathematicaleconomicsandinvexity,respectively.Thencome twenty four full length papers on new achievements in both the theory of the field and its applications. The diapason of the topics tackled in these contributions is very large. It encompasses, in particular, variational inequalities, equilibrium problems, game theory, optimization, control, numerical methods in solving multiobjective optimization problems, consumer preferences, discrete convexity and many others. The volume is a fruit of intensive work of more than hundred specialists all over the world who participated at the latest symposium organized by the Working Group on Generalized Convexity (WGGC) and hosted by the Insubria University. This is the 6th proceedings edited by WGGC, an inter- disciplinary research community of more than 300 members from 36 coun- tries (http://www.gencov.org). We hope that it will be useful for students, VI Preface researchers and practitioners working in applied mathematics and related ar- eas. Acknowledgement. We wish to thank all the authors for their contributions, and all the referees whose hard work was indispensable for us to maintain the scientific quality of the volume and greatly reduce the publica- tion delay. Special thanksgototheInsubriaUniversity fortheorganizational and financial support of the symposium which has contributed greatly to the success of the meeting and its outcome in the form of the present volume. Kazan, Avignon and Ballarat I.V. Konnov August 2006 D.T. Luc A.M. Rubinov Contents Part I Invited Papers Combined Relaxation Methods for Generalized Monotone Variational Inequalities Igor V. Konnov.................................................. 3 Abstract Convexity and the Monge–Kantorovich Duality Vladimir L. Levin................................................ 33 Optimality Conditions and Duality for Multiobjective Programming Involving (C,α,ρ,d) type-I Functions Dehui Yuan, Altannar Chinchuluun, Xiaoling Liu, Panos M. Pardalos .. 73 Part II Contributed Papers Partitionable Variational Inequalities with Multi-valued Mappings Elisabetta Allevi, Adriana Gnudi, Igor V. Konnov.................... 91 Almost Convex Functions: Conjugacy and Duality Radu Ioan Bo¸t, Sorin-Mihai Grad, Gert Wanka......................101 Pseudomonotonicity of a Linear Map on the Interior of the Positive Orthant Alberto Cambini, Laura Martein ...................................115 An Approach to Discrete Convexity and Its Use in an Optimal Fleet Mix Problem Riccardo Cambini, Rossana Riccardi, U¨mit Yu¨ceer ...................133 VIII Contents A Unifying Approach to Solve a Class of Parametrically- Convexifiable Problems Riccardo Cambini, Claudio Sodini..................................149 Mathematical Programming with (Φ,ρ)-invexity Giuseppe Caristi, Massimiliano Ferrara, Anton Stefanescu ............167 Some Classes of Pseudoconvex Fractional Functions via the Charnes-Cooper Transformation Laura Carosi, Laura Martein ......................................177 Equilibrium Problems Via the Palais-Smale Condition Ouayl Chadli, Zaki Chbani, Hassan Riahi ...........................189 Points of Efficiency in Vector Optimization with Increasing- along-rays Property and Minty Variational Inequalities Giovanni P. Crespi, Ivan Ginchev, Matteo Rocca.....................209 HigherOrderProperlyEfficientPointsinVectorOptimization Ivan Ginchev, Angelo Guerraggio, Matteo Rocca .....................227 Higher-order Pseudoconvex Functions Ivan Ginchev, Vsevolod I. Ivanov ..................................247 Sufficient Optimality Conditions and Duality in Nonsmooth Multiobjective Optimization Problems under Generalized Convexity Giorgio Giorgi, Bienvenido Jiménez, Vicente Novo ...................265 Optimality Conditions for Tanaka’s Approximate Solutions in Vector Optimization César Gutiérrez, Bienvenido Jiménez, Vicente Novo..................279 On the Work of W. Oettli in Generalized Convexity and Nonconvex Optimization – a Review and Some Perspectives Joachim Gwinner ................................................297 Local and Global Consumer Preferences Reinhard John...................................................315 Optimality Conditions for Convex Vector Functions by Mollified Derivatives Davide La Torre .................................................327 On Arcwise Connected Convex Multifunctions Davide La Torre .................................................337 A Sequential Method for a Class of Bicriteria Problems Laura Martein, Valerio Bertolucci..................................347 Contents IX Decomposition of the Measure in the Integral Representation of Piecewise Convex Curves Mariana Nedelcheva..............................................359 Rambling Through Local Versions of Generalized Convex Functions and Generalized Monotone Operators Huynh Van Ngai, Jean-Paul Penot .................................379 Monotonicity and Dualities Jean-Paul Penot .................................................399 On Variational-like Inequalities with Generalized Monotone Mappings Vasile Preda, Miruna Beldiman, Anton Ba˘t˘atorescu ..................415 Almost Pure Nash Equilibria in Convex Noncooperative Games Tadeusz Radzik, Wojciech Pol(cid:1)owczuk ...............................433 ASpectralApproachtoSolveBox-constrainedMulti-objective Optimization Problems Maria Cristina Recchioni .........................................449 Part I Invited Papers