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Handbook of Generalized Convexity and Generalized Monotonicity Nonconvex Optimization and Its Applications Volume 76 Handbook of Generalized Convexity and Generalized Monotonicity Nonconvex Optimization and Its Applications Volume 76 Managing Editor: Panos Pardalos University of Florida, U.S.A. Advisory Board: J. R. Birge University of Michigan, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. Floudas Princeton University, U.S.A. J. Mockus Lithuanian Academy of Sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis Technical University Braunschweig, Germany H. Tuy National Centre for Natural Science and Technology, Vietnam Handbook of Generalized Convexity and Generalized Monotonicity Edited by NICOLAS HADJISAVVAS University of Aegean SÁNDOR KOMLÓSI University of Pécs SIEGFRIED SCHAIBLE University of California at Riverside Springer eBook ISBN: 0-387-23393-8 Print ISBN: 0-387-23255-9 ©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: http://ebooks.kluweronline.com and the Springer Global Website Online at: http://www.springeronline.com Contents Preface vii Contributing Authors xvii Part I GENERALIZED CONVEXITY 1 Introduction to Convex and Quasiconvex Analysis 3 Johannes B. G. Frenk, Gábor Kassay 2 Criteria for Generalized Convexity and Generalized Monotonicity 89 in the Differentiable Case Jean-Pierre Crouzeix 3 Continuity and Differentiability of Quasiconvex Functions 121 Jean-Pierre Crouzeix 4 Generalized Convexity and Optimality Conditions in Scalar and 151 Vector Optimization Alberto Cambini and Laura Martein 5 Generalized Convexity in Vector Optimization 195 Dinh The Luc 6 Generalized Convex Duality and Its Economic Applications 237 Juan Enrique Martinez-Legaz 7 Abstract Convexity 293 Alexander Rubinov, Joy deep Dutta vi GENERALIZED CONVEXITY AND MONOTONICITY 8 Fractional Programming 335 Johannes B. G. Frenk, Siegfried Schaible Part II GENERALIZED MONOTONICITY 9 Generalized Monotone Maps 389 Nicolas Hadjisavvas, Siegfried Schaible 10 Generalized Convexity and Generalized Derivatives 421 Sándor Komlósi 11 Generalized Convexity, Generalized Monotonicity and Nonsmooth 465 Analysis Nicolas Hadjisavvas 12 Pseudomonotone Complementarity Problems and Variational Inequalities 501 Jen-Chih Yao, Ouayl Chadli 13 Generalized Monotone Equilibrium Problems and Variational Inequalities 559 Igor Konnov 14 Uses of Generalized Convexity and Generalized Monotonicity in 619 Economics Reinhard John Index 667 Preface Convexity of functions plays a central role in many branches of ap- plied mathematics. One of the reasons is that it is very well fitted to extremum problems. For instance, some necessary conditions for the existence of a minimum become also sufficient in the presence of con- vexity. However, by far not all real-life problems can be described by a convex mathematical model. In many cases, nonconvex functions pro- vide a much more accurate representation of reality. It turns out that often these functions, in spite of being nonconvex, retain some of the nice properties and characteristics of convex functions. For instance, their presence may ensure that necessary conditions for a minimum are also sufficient or that a local minimum is also a global one. This led to the introduction of several generalizations of the classical concept of a convex function. It happened in various fields such as economics, en- gineering, management science, probability theory and various applied sciences for example, mostly during the second half of the last (20th) century. For instance, a well-known and useful property of a convex function is that its sublevel sets are convex. Many simple nonconvex functions also have this property. If we consider the class of all functions whose sublevel sets are convex, we obtain what is called the class of quasiconvex functions. It is much larger than the class of convex functions. The introduction of quasiconvex functions is usually attributed to de Finetti, in 1949. However quasiconvexity played already a crucial role in the 1928 minimax theorem of John von Neumann where it was introduced as a technical hypothesis, not as a new type of a function1. Many other classes of generalized convex functions have been intro- duced since then. In recent years, besides real-valued generalized convex functions, also vector-valued as well as multi-valued generalized convex 1See A. Guerraggio and E. Molho, “The origins of quasi-concavity: a development between mathematics and economics”, Historia Mathematica 31, 2004, 62–75. viii GENERALIZED CONVEXITY AND MONOTONICITY functions have been investigated intensively. During the last forty years, a significant increase of research activities in this field has been wit- nessed. A well-known feature of convexity is that it is closely related to mono- tonicity: a differentiable function is convex if and only if its gradient is a monotone map. This can be extended to nondifferentiable functions as well, through the use of generalized derivatives, subdifferentials and multi-valued maps. Similar connections were discovered between gen- eralized convex functions and certain classes of maps, generically called generalized monotone. For instance, a differentiable function is quasi- convex if and only if its gradient has a property that, quite naturally, was called quasimonotonicity. Generalized monotone maps are not new in the literature. It is very interesting to observe that the first appearance of generalized monotonic- ity (well before the birth of this terminology) dates back to 1936. Even more remarkable, it occurred independently and almost at the same time in two seminal articles: the one, by Georgescu-Roegen (1936), dealt with the concept of a local preference in consumer theory of economics, and the other one, by Wald (1936), contained the first rigorous proof of the existence of a competitive general equilibrium. It is also remarkable that various axioms on revealed preferences in consumer theory are in fact generalized monotonicity conditions. The recognition of close links between the already well established field of generalized convexity and the relatively undeveloped field of general- ized monotonicity gave a boost to both and led to a major increase of research activities. Today generalized monotonicity is frequently used in complementarity problems, variational inequalities and equilibrium problems. The first volume devoted exclusively to generalized convexity were the Proceedings of a NATO Advanced Study Institute organized by Avriel, Schaible and Ziemba in Vancouver, Canada on August 4-15, 1980; “Gen- eralized Concavity in Optimization and Economics” by Schaible and Ziemba (1981, Academic Press). It was followed by the monograph “Generalized Concavity” by Avriel, Diewert, Schaible, Zang (1988, Plenum Publishing Corporation). Both volumes maintain the close re- lationship between the mathematics of generalized convex functions and its relevance to applications, especially in economic theory which de- scribed the field from the beginning. It was Mordecai Avriel who initiated the first monograph solely de- voted to generalizations of convexity back in 1978. Shortly after that he also proposed to organize an international conference on this topic. He PREFACE ix often emphasized the importance of generalizing convexity with applica- tions in mind. This gives the research area its interdisciplinary nature. The extensive amount of new knowledge accumulated since the initial two volumes, exclusively contained in research papers, gave birth to the idea of presenting the main research results in both generalized convexity and generalized monotonicity in the form of a handbook. The present volume consists of fourteen chapters, written by lead- ing experts of various areas of Generalized Convexity and Generalized Monotonicity. Each chapter starts from the very basics and goes to the state of the art of its subject. Given the amount of material in the field, there are some topics that were not included. We trust that the com- prehensive bibliography provided by each chapter can help to bridge the gap. The chapters are grouped into two parts of the book, depending on whether their main focus is generalized convexity or generalized mono- tonicity. Chapter 1, by Frenk and Kassay, provides an introduction to con- vex and quasiconvex analysis in a finite-dimensional setting. Since this analysis is heavily based on the study of certain sets, the most impor- tant algebraic and topological properties of linear subspaces, affine sets, convex and evenly convex sets and cones are included here. The well- known separation results are also presented and then used to derive the so-called dual representations for (evenly) convex sets. The next part is devoted fundamentally to the study of convex, quasiconvex and evenly quasiconvex functions. It is shown that this study can be reduced to the study of the sets considered in the preceding part. As such, the equiva- lent formulation of a separation result for convex sets is now given by the dual representation of a function. The key result in this respect is the Fenchel-Moreau theorem within convex analysis and its generalization to evenly quasiconvex functions. The remainder of the chapter presents some important applications of convex and quasiconvex analysis to opti- mization theory,
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