Vector Variational Inequalities and Vector Equilibria Nonconvex Optimization and Its Applications Volume 38 Managing Editors: Panos Pardalos University of Florida, U.S.A. Reiner Horst University of Trier, Germany Advisory Board: 1. R. Birge University of Michigan, U.S.A. Ding-Zhu Du University of Minnesota, U.S.A. C. A. F10udas Princeton University, U.S.A. J. Mockus Stanford University, U.S.A. H. D. Sherali Virginia Polytechnic Institute and State University, U.S.A. G. Stavroulakis Technical University Braunschweig, Germany The titles published in this series are listed at the end of this volume. Vector Variational Inequalities and Vector Equilibria Mathematical Theories Edited by Franco Giannessi Department of Mathematics, University of Pisa, Pisa, Italy KLUWER ACADEMIC PUBLISHERS DORDRECHT/BOSTON/LONDON A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13:978-1-4613-7985 -0 e-ISBN -13: 978-1-4613-0299·5 DOl: 10.1007/978-1-4613-0299·5 Published by Kluwer Academic Publishers. P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved © 2000 Kluwer Academic Publishers Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner CONTENTS Preface ....................................................... xiii Vector Equilibrium Problems and Vector Variational Inequalities ............................................... pag. 1 A.H. Ansari 1. Introduction ............................................ pag. 1 2. Preliminaries ............................................ pag. 3 3. Existence Results ........................................ pag. 4 4. Strongly Nonlinear Vector Variational Inequalities ....... pag. 12 References ................................................ pag. 14 Generalized Vector Variational-Like Inequalities and their Scalarization .. ............................... pag. 17 A.H. Ansari, A.H. Siddiqi and J.-C. Yao 1. Introduction ........................................... pag. 18 2. Preliminaries .......................................... pag. 20 3. Existence results ........................................ pag. 23 4. Scalarization ........................................... pag. 31 Reference ................................................. pag. 34 Existence of Solutions for Generalized Vector Variational-Like Inequalities .......................... pag. 39 S.-S. Chang, H.B. Thompson and G.X-Z. Yuan 1. Introduction and Preliminaries .......................... pag. 40 2. Existence Theorems .................................... pag. 43 References ................................................ pag. 51 On Gap Functions for Vector Variational Inequalities .............................................. pag. 55 G. - Y. Chen, C. --J. Goh and X. Q. Yang 1. Introduction ............................................ pag. 56 2. Mathematical Preliminaries ............................. pag. 56 3. A First Gap Function ................................... pag. 59 4. Convexity of the First Gap Function .................... pag. 64 VI 5. A Second Gap Function ................................ pag. 67 6. Concluding Remarks .................................... pag. 70 References ................................................ pag. 70 Existence of Solutions for Vector Variational Inequalities .............................................. pag. 73 G.-Y. Chen and S.-H. Hou 1. Introduction ............................................ pag. 73 2. Models and Basic Results ............................... pag. 74 3. A Kind of Generalized Vector Variational-Like Inequalities ............................................. pag. 79 4. Comments .............................................. pag. 82 References ................................................ pag. 83 On the Existence of Solutions to Vector Complementarity Problems ............................ pag. 87 G. -Y. Chen and X. Q. Yang 1. Introduction ............................................ pag. 87 2. Vector Complementarity Problems ...................... pag. 89 3. Vector Implicit Complementarity Problems ............. pag. 91 4. Generalized Vector Complementarity Problems .......... pag. 92 5. Conclusions ............................................ pag. 94 References ................................................ pag. 95 Vector Variational Inequalities and Modelling of a Continuum Traffic Equilibrium Problem ............ pag. 97 P. Daniele and A. M augen 1. Introduction ............................................ pag. 97 2. A new model of Traffic Equilibrium Problem in the Continuous Case ....................................... pag. 99 3. A General Continuous Problem ........................ pag. 104 4. Lagrangean Theory .................................... pag. 104 5. A Computational Procedure ........................... pag. 108 References ............................................... pag. 109 Generalized Vector Variationa-Like Inequalities without Monotonicity ................................. pag. 113 x.P. Ding and E. Tarafdar Vll 1. Introduction ........................................... pag. 113 2. Existence of Solutions ................................. pag. 117 References ............................................... pag. 123 Generalized Vector Variationa-Like Inequalities with C -7]-Pseudomonotone Set-Valued Mappings ...... pag. 125 x x.P. Ding and E. Tara/dar 1. Introduction ........................................... pag. 126 2. Preliminaries .......................................... pag. 126 3. Existence Theorems ................................... pag. 132 References ............................................... pag. 139 A Vector Variationa-Like Inequality for Compact Acyclic Multifunctions and its Applications ........ pag. 141 J. Fu 1. Introduction ........................................... pag. 141 2. Preliminaries .......................................... pag. 142 3. Vector Variational-Like Inequality ..................... pag. 144 4. GVQVI and GVQCP .................................. pag. 148 References ............................................... pag. 150 On the Theory of Vector Optimization and Variational Inequalities. Image Space Analysis and Separation ....................................... pag. 153 F. Giannessi, G. Mastroeni and L. Pellegrini 1. Introduction ........................................... pag. 153 2. Image Space and Separation for VOP .................. pag. 154 3. Other Kinds of Separation ............................. pag. 164 4. Separation in the Weak Case .......................... pag. 169 5. Necessary Optimality Conditions ...................... pag. 171 6. Saddle Point Conditions ............................... pag. 177 7. Duality ................................................ pag. 181 8. Scalarization of Vector Optimization ................... pag. 187 9. Image Space and Separation for VVI ................... pag. 197 10. Scalarization of VVI. ................................. pag. 202 11. Some Remarks on Penalization ....................... pag. 208 References .............................. , ................ pag. 211 Vlll Scalarization Methods for Vector Variational Inequality .............................................. pag. 217 C.J. Goh and X.Q. Yang 1. Introduction ........................................... pag. 217 2. Preliminary Results ................................... pag. 219 3. Scalarization .......................................... pag. 222 4. Affine WVVI. An Active-Set Method ................. pag. 226 5. Conclusion ............................................ pag. 231 References ............................................... pag. 231 Super Efficiency for a Vector Equilibrium in Locally Convex Topological Vector Spaces ................... pag. 233 x.H. Gong, W. T. Fu and W. Liu 1. Introduction ........................................... pag. 233 2. Super Efficiency and Scalarization ..................... pag. 237 3. Connectedness ......................................... pag. 243 References ............................................... pag. 250 The Existence of Essentially Connected Components of Solutions for Variational Inequalities ............ pag. 253 G. Isac and G.X.Z. Yuan 1. Introduction ........................................... pag. 254 2. The Upper Semicontinuity of Solution set for Variational Inequalities ........................................... pag. 256 3. The Existence of Essentially Connected Components for VI with Set-Valued Mappings ...................... pag. 259 References ............................................... pag. 264 Existence of Solutions for Vector Saddle-Point Problems ............................................... pag. 267 K.R. Kazmi 1. Introduction ........................................... pag. 267 2. Existence of Solutions ................................. pag. 269 References ............................................... pag. 274 Vector Variational Inequality as a Tool for Studying Vector Optimization Problems ....................... pag. 277 IX C.M. Lee, D.S. Kim, B.S. Lee and N.D. Yen 1. Introduction ........................................... pag. 277 2. Vector Variational Inequality .......................... pag. 278 3. Vector Variational Inequalities and Vector Optimization Problems .............................................. pag. 281 4. Strongly Monotone Vector Variational Inequalities ..... pag. 286 5. Sensitivity of the Solution set of a Perturbed Strongly Monotone VVI ........................................ pag. 293 6. Example .............................................. pag. 300 References ............................................... pag. 304 Vector Variational Inequalities in a Hausdorff Topological Vector Space ............................. pag. 307 C.M. Lee and S. Kum 1. Introduction ........................................... pag. 307 2. Preliminaries .......................................... pag. 308 3. Two Existence Results ................................. pag. 310 4. Generalized Minty Vector Variational Inequality ....... pag. 315 References ............... , .................... , .......... pag. 318 Vector Ekeland Variational Principle ................ pag. 321 S.l. Li, XQ. Yang and C.-Y. Chen 1. Introduction ........................................... pag. 321 2. Vector Ekeland Variational Principle I ................. pag. 322 3. Vector Ekeland Variational Principle II ................ pag. 328 4. Conclusions ........................................... pag. 332 References ............................................... pag. 332 Convergence of Approximate Solutions and Values in Parametric Vector Optimization .................... pag.335 P. Loridan and l. Morgan 1. Introduction ........................................... pag. 335 2. Minimal and to-Minimal Solution ....................... pag. 336 3. Convergence of Parametric Minimal and Approximate Values ................................................ pag. 339 4. Convergence of Restricted to-Minimal Solutions ......... pag. 345 References ............................................... pag. 347