Generalized Convexity: Proceedings of the IVth International Workshop on Generalized Convexity Held at Janus Pannonius University Pécs, Hungary, August 31–September 2, 1992 PDF
Preview Generalized Convexity: Proceedings of the IVth International Workshop on Generalized Convexity Held at Janus Pannonius University Pécs, Hungary, August 31–September 2, 1992
Lecture Notes in Economics and Mathematical Systems 405 Founding Editors: M. Beckmann H. P. Kiinzi Editorial Board: H. Albach, M. Beckmann, O. Feichtinger, W. Hildenbrand, W. Krelle H. P. Kiinzi, K. Ritter, U. Schittko, P. Schonfeld, R. Selten Managing Editors: Prof. Dr. O. Fandel Fachbereich Wirtschaftswissenschaften Femuniversitat Hagen Feithstr. 140/AVZ II, D-58097 Hagen, FRO Prof. Dr. W. Trockel Institut fUr Mathematische Wirtschaftsforschung (IMW) Universitat Bielefeld Universitatsstr. 25, D-33615 Bielefeld, FRO S. Koml6si T. Rapcsak S. Schaible (Eds.) Generalized Convexity Proceedings of the IVt h International Workshop on Generalized Convexity Held at Janus Pannonius University Pecs, Hungary, August 31-September 2, 1992 Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Editors Prof. Dr. Sandor Koml6si Faculty of Economics, Janus Pannonius University Rak6czi 6t 80, H-7621 Pecs, Hungary Prof. Dr. Tamas Rapcsak Computer and Automation Institute Hungary Academy of Sciences P.O. Box 63, Kende u. 13-17, H-1518 Budapest, Hungary Prof. Dr. Siegfried Schaible Graduate School of Management University of California Riverside, CA 92521, USA ISBN-13: 978-3-540-57624-2 e-ISBN-13: 978-3-642-46802-5 001: 10.1007/978-3-642-46802-5 Library of Congress Cataloging-in-Publication Data. International Workshop on Generalized Convexity (4th: 1992: Pecs, Hungary) Generalized convexity: pro ceedings of the Fourth International Workshop on Generalized Convexity, held in Pecs, Hungary, August 31-September 2, 19921 [edited by] S. Koml6si, T. Rapcsak, S. Schaible. p. cm. - (Lecture notes in economics and mathematical systems; 405) ISBN-13: 978-3-540-57624-2 1. Convex functions-Congresses I. Koml6si, S. (Sandor), 1947- ll. Rapczak, T. (Tamas), 1947-ill. Schaible, Siegfried. IV. TItle. V. Series. QA301.5.159 1994 515'.88-dc20 93-44607 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Typesetting: Camera ready by author SPIN: 10083791 4213140-543210 -Printed on acid-free paper Preface Generalizations of the classical concept of a convex function have been pro posed in various fields such as economics, management science, engineering, statistics and applied sciences during the second half of this century. The present volume constitutes the proceedings of the Fourth Interna tional Workshop on Generalized Convexity in Pecs, Hungary, August 31 - September 2, 1992. The proceedings are edited by the organizers of the work shop, Sandor Koml6si, Janus Pannonius University, Pecs, Tamas Rapcsak, Hungarian Academy of Sciences, Budapest and Siegfried Schaible, University of California, Riverside. Papers at the conference were carefully refereed and a selection of them is published herewith. Thanks are due to all referees for their generous and prompt help. Previous conferences on generalized convexity were held in Vancouver in 1980 (organized by M. Avriel, S. Schaible, W. T. Ziemba), in Canton in 1986 (organized by C. Singh) and in Pisa in 1988 (organized by A. Cambini, E. Castagnoli, 1. Martein, P. Mazzoleni, S. Schaible). Like its predecessors, this fourth conference was a truely international event with 76 participants from 17 countries. We were pleased that Bela Martos, Budapest known for his early contributions to this field, served as honorary chairman of the conference. In addition to new results in more established areas of generalized con vexity, several important developments in recently emerging areas were pre sented. Also, a number of interesting applications were reported. We wish to express our sincere gratitude to the leaders of Janus Pannonius University, in particular Professor J6zsef Voros, Dean of the Faculty of Eco nomics for their outstanding support of this conference. Furthermore, we are deeply indebted to the National Scientific Research Foundation (OTKA), the Foundation for Raising the Hungarian Economists Training (MHB RT.), the Illyes Foundation and MALEV, The Hungarian Airlines Company for their generous support. Special thanks are due to Peter Dombi for his excellent work in typesetting all the manuscripts. Finally we thank Dr. Werner A. Miiller, Springer-Verlag for his outstand ing help in the production of this volume. Contents Part I. Generalized convex functions Co Ro BECTOR, So CHANDRA, So GHUPTA, So K. SUNEJA: Univex sets, functions and univex nonlinear programming 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 Lo BLAGA, Jo KOLUMBAN: Optimization on closely convex sets 00000 19 Mo CIGOLA: A note on ordinal concavity 00000000000000000000000000000 35 THo DRIESSEN: Generalized concavity in cooperative game theory: characterizations in terms of the core 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 40 Fo FORGO: On the existence of Nash-equilibrium in n-person generalized concave games 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 53 Jo Bo Go FRENK, Jo GROMICHO, Fo PLASTRIA, So ZHANG: A deep cut ellipsoid algorithm and quasiconvex programming 62 0 0 0 00 000 0 0 0 000 Ho HARTWIG: Quasiconvexity and related properties in the calculus of variations 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 77 Jo Ao MAYOR-GALLEGO, Ao RUFIAN-LIZANA, Po RUIZ-CANALES: Ray-quasiconvex and f-quasiconvex functions 0000000000000000000000 85 To RAPcsAK: Geodesic convexity on IR" 00000000000000000000000000000 91 Po SZILAGYI: A class of differentiable generalized convex functions 0000104 Mo TOSQUES: Equivalence between generalized gradients and sub differentials (lower semigradients) for a suitable class of lower semicontinuous functions 116 000000000000000000000000000000000000000000 Part II. Optimality and duality I. Ho BOMZE, Go DANNINGER: Generalizing convexity for second order optimali ty conditions 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 137 Po Ho DIEN, Go MASTROENI, Mo PAPPALARDO, Po Ho QUANG: Regularity conditions for constrained extremum problems via image space approach: the linear case 000000000000000000000000000000000000 145 Jo Go Bo FRENK, Do Mo Lo DIAS, Jo GROMICHO: Duality theory for . convex/quasiconvex functions and its application to optimization 000153 Go GIORGI, Ao GUERRAGGIO: First order generalized optimality conditions for programming problems with a set constraint 0 0 0 0 0 0 0 0 0 171 Vlll Contents B. M. GLOVER, V. JEYAKUMAR: Abstract nonsmooth nonconvex programming ...................................................... 186 S. MITITELU: A survey on optimality and duality in nonsmooth programming ...................................................... 211 Part III. Generalized monotone maps S. SCHAIBLE: Generalized monotonicity - a survey ................... 229 E. CASTAGNOLI, P. MAZZOLENI: Orderings, generalized convexity and monotonicity .................................................. 250 S. KOMLOSI: Generalized monotonicity in non-smooth analysis ....... 263 R. PINI, S. SCHAIBLE: Some invariance properties of generalized monotonicity ...................................................... 276 Part IV. Fractional programming I. A. BYKADOROV: On quasiconvexity in fractional programming .... 281 R. CAMBINI: A class of non-linear programs: theoretical and algorithmical results ............. , ................................. 294 A. CSEBFALVI, G. CSEBFALVI: Post-buckling analysis of frames by a hybrid path-following method ................................ 311 I. M. STANCU-MINASIAN, S. TIGAN: Fractional programming under uncertainty ........................................................ 322 Part V. Multiobjective programming A. CAMBINI, L. MARTEIN: Generalized concavity and optimality conditions in vector and scalar optimization ....................... 337 C. R. BECTOR, M. K. BECTOR, A. GILL, C. SINGH: Duality for vector valued B-invex programming ............................... 358 J. FULOP: A cutting plane algorithm for linear optimization over the efficient set ........................................................ 374 H. ISHII: Multiobjective scheduling problems ......................... 386 A. MARCHI: On the relationships between bicriteria problems and non-linear programming ........................................... 392 Contributing authors ................................................. 401 PART I. GENERALIZED CONVEX FUNCTIONS U nivex sets, functions and univex nonlinear programming C. R. Bector, S. Chandra, S. Gupta and S. K. Suneja Faculty of Management, University of Manitoba, Winnipeg, Canada Department of Mathematics, Indian Institute of Technology, New Delhi, India Department of Mathematics, Delhi University, Delhi, India Department of Mathematics, Delhi University, Delhi, India In the present paper we introduce the concept of univex sets, and define a new class of functions, called univex functions, on them. These functions unify the concepts of convexity, B-vexity, invexity and B-invexity. Some of their properties are proved and applications in nonlinear programming are discussed. Furthermore, generalized univex functions are also introduced and their relationships to univex functions and convex (generalized convex) functions are also discussed. Under appropiate assumptions of univexity, op timality conditions and duality results for Mond-Weir duality are established. In the end some suggestions for further research have been made. 1. Introduction The class of B-vex functions has been recently introduced by Bector and Singh [2] as a generalization of convex functions. Similar functions were introduced by Bector [1] and Castagnoli and Mazoletti [5]. The concept of convexity of functions was generalized to invex functions by Hanson [6], and to preinvex functions by Ben Israel and Mond [4], respectively, which were further generalized to B-invex functions by Bector, Suneja and Lalitha [3] and to B-preinvex functions by Suneja, Singh and Bector [10]. In the present paper, the concepts of univex sets and a new class of func tions, called univex functions, are introduced by generalizing the concepts of convexity, B-vexity, invexity, and B-invexity. Certain properties of uni vex functions in terms of univex sets are established and their relations with convex, B-vex, invex, and B-invex functions are established. Univex func tions are further extended to pseudounivex and quasiunivex functions. To show their applications, sufficient optimality conditions and duality results for Mond-Weir duality [9] are established for a nonlinear programming prob lem involving univex functions. In the end some generalizations for V-invex functions, recently introduced by Jeyakumar and Mond [7] are suggested. Furthermore, some suggestions for further research have also been made. 4 c. R. Bector et al. 2. Definitions and some properties Let X ~ JRn be non empty, 7J : X x X -+ JRn b : X x X x [0,1] -+ JR+, f : X -+ lR and 41 : JR -+ JR. For x E X, u E X, 0::; ..\ ::; 1, we assume that b stands for b(x, u,..\) ~ 0, and ..\b ::; l. Definition 2.1 A functional f : lRn -+ lR is said to be (i) increasing if and only if x::; y => f(x)::; f(y), x, y E lRn . (ii) strictly increasing if and only if x < y => f(x) < f(y), x, y E JRn . Definition 2.2 A functional f : JRn -+ lR is said to be (i) decreasing if and only if x::; y => f(x) ~ f(y), X,y E JRn . (ii) strictly decreasing if and only if x < y => f(x) > f(y), X,y E JRn . Definition 2.3 A function f is said to be sublinear over a space S if (A) f(x + y) ::; f(x) + f(y) "Ix, y E 5, (B) f(ax) = af(x) a E JR, a ~ 0, x E 5 . Definition 2.4 A function f is said to be superlinear over a space S if (A) f(x + y) ~ f(x) + f(y) "Ix, y E 5, (B) f(ax) = af(x) a E JR, a ~ 0, x E 5 . Remark 2.1 From the definitions it follows that whenever f is a sublinear or superlinear function over a space 5, f(O) = o. Definition 2.5 (Bector and Singh [2]). At u E X, where X is a convex set, a function f is said to be B-vex with respect to b if for every x E X and 0::;,,\::;1 f[..\x + (1 - ..\)u] < ..\b(x, u, ..\)f(x) + (1 - ..\b(x, u, ..\))f(u) f(u) + ..\b(x, u, ..\)[f(x) - f(u)] . Univex sets, functions and univex nonlinear programming 5 Definition 2.6 Let u E X. The set X is said to be invex at u with + respect to '1 if for each x E X and 0 ~ A ~ 1, u A'1(X, u) EX. X is said to be an invex set with respect to '1 if X is invex at each x EX. Definition 2.7 At u E X, where X is an invex set, the function f is said to be pre-univex (pre-UVX) (strictly pre-UVX) with respect to '1, ~ and b, if for every z E X and 0 ~ A ~ I, f[u + A'1(Z,U)] ~ «) f(u) + Ab~[f(z) - f(u)] . Definition 2.8 At u E X, where X is an invex set, the function f is said to be pre-quasiunivex (pre-QUVX) with respect to '1, ~ and b, if for every z E X and 0 ~ ~ ~ 1, ~[f(z) - f(u)] ~ 0 ~ bf[u + ~'1(z, u)] ~ bf(u) . Definition 2.9 At u EX, where X is an in vex set, a function f is said to be strongly pre-quasiunivex (S. pre-QUVX) with respect to '1, ~ and b, if for every z EX, z :/; u and 0 < ~ ~ 1, ~[f(z) - f(u)] ~ 0 ~ bf[u + ~'1(z, u)] < bf(u) . Definition 2.10 Given S ~ JRn x JR. S is said to be a univex set with respect to '1, ~ and b, if every (z,O'),(u,p)ESandO~~~l ~ (u+~'1(z,u),P+Ab~(O'-P»ES. Let f be a differentiable function defined on a nonempty subset of JRn and let ~ : JR - JR and b : X x X - 1R+. For z EX, u EX, we write b(z,u) = l~b(z,u,A) ~ O. In what follows while using them we shall make no distinction between b( x, u) and b(z, u, A) and write b for either of them. Definition 2.11 (Hanson [6], Ben Israel and Mond [4].) At u E X, the function f is said to be invex with respect to '1, if for every x EX, f(z)-f(u) ~ '1(z,u)TVf(u).
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