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Studies in generalized convexity and generalized monotonicity have significantly increased during the last two decades. Researchers with very diverse backgrounds such as mathematical programming, optimization theory, convex analysis, nonlinear analysis, nonsmooth analysis, linear algebra, probability theory, variational inequalities, game theory, economic theory, engineering, management science, equilibrium analysis, for example are attracted to this fast growing field of study. Such enormous research activity is partially due to the discovery of a rich, elegant and deep theory which provides a basis for interesting existing and potential applications in different disciplines. The handbook offers an advanced and broad overview of the current state of the field. It contains fourteen chapters written by the leading experts on the respective subject; eight on generalized convexity and the remaining six on generalized monotonicity.
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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: Print ISBN: 0-387-23393-8 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: and the Springer Global Website Online at: http://ebooks.kluweronline.com http://www.springeronline.com Contents vii Preface Contributing Authors Part I xvii GENERALIZED CONVEXITY 1 Introduction to Convex and Quasiconvex Analysis Johannes B. G. Frenk, Gábor Kassay 2 Criteria for Generalized Convexity and Generalized Monotonicity in the Differentiable Case Jean-Pierre Crouzeix 3 Continuity and Differentiability of Quasiconvex Functions Jean-Pierre Crouzeix 4 Generalized Convexity and Optimality Conditions in Scalar and Vector Optimization Alberto Cambini and Laura Martein 3 89 121 151 5 Generalized Convexity in Vector Optimization Dinh The Luc 195 6 Generalized Convex Duality and Its Economic Applications Juan Enrique Martinez-Legaz 237 7 Abstract Convexity Alexander Rubinov, Joy deep Dutta 293 vi GENERALIZED CONVEXITY AND MONOTONICITY 8 Fractional Programming Johannes B. G. Frenk, Siegfried Schaible 335 Part II GENERALIZED MONOTONICITY 9 Generalized Monotone Maps Nicolas Hadjisavvas, Siegfried Schaible 389 10 Generalized Convexity and Generalized Derivatives Sándor Komlósi 421 11 Generalized Convexity, Generalized Monotonicity and Nonsmooth Analysis Nicolas Hadjisavvas 465 12 Pseudomonotone Complementarity Problems and Variational Inequalities Jen-Chih Yao, Ouayl Chadli 501 13 Generalized Monotone Equilibrium Problems and Variational Inequalities Ig