Equilibrium Problems and Variational Models

Transcription

1 Equilibrium Problems and Variational Models

2 Nonconvex Optimization and Its Applications Volume 68 Managing Editor: Panos Pardalos University 0/ Florida, U.SA. Advisory Board: J. R. Birge University o/michigan, U.SA. Ding-ZhuDu University o/minnesota, U.SA. c. A. Floudas Princeton University, U.SA. J. Mockus Lithuanian Academy o/sciences, Lithuania H. D. Sherali Virginia Polytechnic Institute and State University, U.SA. G. Stavroulakis Technical University Braunschweig, Germany

3 Equilibrium Problems and Variational Models Edited by Patrizia Daniele Department of Mathematics University of Catania Catania Italy Franco Giannessi Department of Mathematics University of Pi sa Pisa Italy Antonino Mangeri Department of Mathematics University of Catania Catania Italy

4 Distributors for North, Central and South America: K1uwer Academic Publishers 101 Philip Drive Assinippi Park Norwell, Massachusetts USA Telephone (781) Fax (781) Distributors for all other countries: K1uwer Academic Publishers Group Post Office Box AlI Dordrecht, THE NETHERLANDS Telephone * Fax <orderdept@wkap.nl> Electronic Services < Library of Congress Cataloging-in-Publication ClPinfo or: Title: Equilibrium Problems and Variational Models Author: Patrizia Daniele, Franco Giannessi, Antonino Maugeri ISBN-13: e-isbn-13: DOl: / Copyright 2003 by K1uwer Academic Publishers Softcover reprint of the hardcover 1st edition 2003 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photo-copying, microfilming, recording, or otherwise, without the prior written permission of the publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Permissions for books published in the USA: perm; ss; oos@wkap com Permissions for books published in Europe: permissions@wkap.nl Printed on acid-free paper.

5 Contents Preface xiii On Vector Quasi-Equilibrium Problems Qamrul Hasan Ansari and Jen-Chih Yao Preliminaries 3 3. Existence Results 6 4. Some Applications The Log-Quadratic Proximal Methodology in Convex Optimization Algorithms and Variational Inequalities 19 Alfred Auslender and Marc Teboulle Lagrangians and Proximal Methods The quadratic augmented Lagrangian Proximal Minimization Algorithms Entropic Proximal Methods and Modified Lagrangians Difficulties with Entropic Proximal Methods Toward Solutions to Difficulties The Logarithmic-Quadratic Proximal Framework The LQ-Function and its Conjugate: Basic Properties The Logarithmic-Quadratic Proximal Minimization The LQP in Action Primal LQP for Variational Inequalities over Polyhedra Lagrangian Methods for convex optimization and variational inequalities Dual and Primal-Dual Decomposition schemes Primal Decomposition: Block Gauss-Seidel Schemes for Linearly constrained Problems 39

6 vi 4.5. Convex Feasibility Problems 4.6. Bundle Methods in Nonsmooth Optimization The Continuum Model of Transportation Problem Patrizia Daniele, Giovanna Idone and Antonino Maugeri 2. Calculus of the solution The Economic Model for Demand-Supply Problems Patrizia Daniele and Antonino Maugeri The first phase: formalization of the equilibrium The second phase: formalization of the equilibrium The dependence of the second phase on the first one General model Example Constrained Problems of Calculus of Variations Via Penalization Technique 79 Vladimir F. Demyanov 2. Statement of the problem 3. An equivalent statement of the problem 4. Local minima 5. Penalty functions 6. Exact penalty functions 6.1. Properties of the function cp 6.2. Properties of the function G 6.3. The rate of descent of the function cp 6.4. An Exact Penalty function 7. Necessary conditions for an Extremum 7.1. Necessary conditions generated by classical variations 7.2. Discussion and Remarks

7 Variational Problems with Constraints Involving Higher- Order Derivatives 109 Vladimir F. Demyanov and Franco Giannessi Statement of the problem An equivalent statement of the problem Local minima Properties of the function 'P A classical variation of z The case z if- Z The case z E Z Exact penalty functions Properties of the function G An Exact Penalty function Necessary conditions for an Extremum vii On the strong solvability of a unilateral boundary value problem for Nonlinear Parabolic Operators in the Plane 135 Rosalba Di Vincenzo Hypotheses and results Preliminary results Proof of the theorems Solving a Special Class of Discrete Optimal Control Problems Via a Parallel Interior-Point Method 141 Carla Dumzzi, Valeria Ruggiero and Gaetano Zanghimti 2. Framework of the Method 3. Global convergence 4. A special class of discrete optimal control problems 5. Numerical experiments 6. Conclusions

8 viii Solving Large Scale Fixed Charge Network Flow Problems 163 Burak Ek iofilu, Sandra Duni Ek iofilu and Panos M. Pardalos Problem Definition and Formulation Solution Procedure The DSSP Local Search Computational Results Concluding Remarks Variable Projection Methods for Large-Scale Quadratic Optimization in data Analysis Applications 185 Emanuele Galligani, Valeria Ruggiero and Luca Zanni Large QP Problems in Training Support Vector Machines Numerical Solution of Image Restoration Problem A Bivariate Interpolation Problem Conclusions Strong solvability of boundary value problems in elasticity with Unilateral Constraints 213 Sofia Giuffre Basic assumptions and main results Preliminary results Proof of the theorems Time Dependent Variational Inequalities - Some Recent Trends 225 Joachim Gwinner 2. Time - an additional parameter in variational inequalities

9 2.1. Time-dependent variational inequalities and quasi-variational inequalities Some classic results on the differentiability of the projection on closed convex subsets in Hilbert space Time-dependent variational inequalities with memory terms Ordinary Differential Inclusions with Convex Constraints: Sweeping Processes Moving convex sets and systems with hysteresis Sweeping processes and generalizations Projected dynamical systems Differentiability of the projection onto closed convex subsets revisited Projected dynamical systems and stationarity Well-posedness for projected dynamical systems Some Asymptotic Results Some classical results An asymptotic result for full discretization Some convergence results for continuous-time subgradient procedures for convex optimization ix On the Contractibility of the Efficient and Weakly Efficient Sets in R2 265 Nguyen Quang Huy, Ta Duy Phuong and Nguyen Dong Yen Preliminaries Topological structure of the efficient sets of compact convex sets Example Existence Theorems for a Class of Variational Inequalities and Applications to a Continuous Model of Transportation 281 Francesco Marino 2. Continuous transportation model 3. Existence Theorem

10 x On Auxiliary Principle for Equilibrium Problems Giandomenico Mastroeni The auxiliary equilibrium problem The auxiliary problem principle Applications to variational inequalities and optimization problems Concluding remarks Multicriteria Spatial Price Networks: Statics and Dynamics Anna Nagurney, June Dong and Ding Zhang 2. The Multicriteria Spatial Price Model 3. Qualitative Properties 4. The Dynamics 5. The Discrete-Time Algorithm 6. Numerical Examples 7. Summary and Conclusions Non regular data in unilateral variational problems Pirro Oppezzi The approach by truncation and approximation Renormalized formulation Multivalued operators and more general measures Uniqueness and convergence Equilibrium Concepts in Transportation Networks: Generalized Wardrop Conditions and Variational Formulations 333 Massimo Pappalardo and Mauro Passacantando 2. Equilibrium model in a traffic network

11 xi 344 Variational Geometry and Equilibrium Michael Patriksson and R. Tyrrell Rockafellar Variational Inequalities and Normals to Convex Sets Quasi-Variational Inequalities and Normals to General Sets Calculus and Solution Perturbations Application to an Equilibrium Model with Aggregation On the Calculation of Equilibrium in Time Dependent Traffic Networks 369 Fabio Raciti 2. Calculation of Equilibria 3. The algorithm 4. Applications and Examples 5. Conclusions Mechanical Equilibrium and Equilibrium Systems Tamas Rapcsak Physical motivation Statement of the mechanical force equilibrium problem The principle of virtual work Characterization of the constraints Quasi-variational inequalities (QVI) Principle of virtual work in force fields under scleronomic and holonomic constraints Dual form of the principle of virtual work in force field under scleronomic and holonomic constraints Procedure for solving mechanical equilibrium problems Existence of solutions

12 xii False Numerical Convergence in Some Generalized Newton Methods 401 Stephen M. Robinson 2. Some generalized Newton methods 3. False numerical convergence 4. An example 5. Avoiding false numerical convergence Distance to the Solution Set of an Inequality with an Increasing Function Alex M. Rubinov Preliminaries Distance to the solution set of the inequality with an arbitrary increasing function Distance to the solution set of the inequality with an ICAR function Inequalities with an increasing function defined on the entire space Inequalities with a topical function Transportation Networks with Capacity Constraints 433 Laum Scrimali 2. Wardrop's generalized equilibrium condition 3. A triangular network 4. More about generalized equilibrium principle 5. Capacity constraints and paradox

13 Preface The volume, devoted to variational analysis and its applications, collects selected and refereed contributions, which provide an outline of the field. The meeting of the title "Equilibrium Problems and Variational Models", which was held in Erice (Sicily) in the period June 23 - July , was the occasion of the presentation of some of these papers; other results are a consequence of a fruitful and constructive atmosphere created during the meeting. New results, which enlarge the field of application of variational analysis, are presented in the book; they deal with the vectorial analysis, time dependent variational analysis, exact penalization, high order derivatives, geometric aspects, distance functions and log-quadratic proximal methodology. The new theoretical results allow one to improve in a remarkable way the study of significant problems arising from the applied sciences, as continuum model of transportation, unilateral problems, multicriteria spatial price models, network equilibrium problems and many others. As noted in the previous book "Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models", edited by F. Giannessi, A. Maugeri and P.M. Pardalos, Kluwer Academic Publishers, Vol. 58 (2001), the progress obtained by variational analysis has permitted to handle problems whose equilibrium conditions are not obtained by the minimization of a functional. These problems obey a more realistic equilibrium condition expressed by a generalized orthogonality (complementarity) condition, which enriches our knowledge of the equilibrium behaviour. Also this volume presents important examples of this formulation. Finally, recent numerical methods complete this volume that, like the previous one, aims to treat problems from their statement up to the computation of their solutions. Patrizia Daniele, University of Catania Franco Giannessi, University of Pisa Antonino Maugeri, University of Catania