Equilibrium Problems and Variational Models

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

Equilibrium Problems and Variational Models

Edited by

Patrizia Daniele Department of Mathematics

University of Catania 95125 Catania

Italy

Franco Giannessi Department of Mathematics

University of Pi sa 56127 Pisa

Italy

Antonino Mangeri Department of Mathematics

University of Catania 95125 Catania

Italy

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Library of Congress Cataloging-in-Publication

ClPinfo or: Title: Equilibrium Problems and Variational Models Author: Patrizia Daniele, Franco Giannessi, Antonino Maugeri ISBN-13:978-1-4613-7955-3 e-ISBN-13:978-1-4613-0239-1 DOl: 10.1007/978-1-4613-0239-1

Copyright © 2003 by K1uwer Academic Publishers Softcover reprint of the hardcover 1st edition 2003

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Contents

Preface

On Vector Quasi-Equilibrium Problems Qamrul Hasan Ansari and Jen-Chih Yao

xiii

1

1. Introduction 1 2. Preliminaries 3 3. Existence Results 6 4. Some Applications 10 References 15

The Log-Quadratic Proximal Methodology in Convex Op-timization Algorithms and Variational Inequalities 19

Alfred Auslender and Marc Teboulle

1. Introduction 20 2. Lagrangians and Proximal Methods 21

2.1. The quadratic augmented Lagrangian 21 2.2. Proximal Minimization Algorithms 22 2.3. Entropic Proximal Methods and Modified Lagrangians 24 2.4. Difficulties with Entropic Proximal Methods 26 2.5. Toward Solutions to Difficulties 28

3. The Logarithmic-Quadratic Proximal Framework 29 3.1. The LQ-Function and its Conjugate: Basic Properties 29 3.2. The Logarithmic-Quadratic Proximal Minimization 31

4. The LQP in Action 33 4.1. Primal LQP for Variational Inequalities over Polyhedra 33 4.2. Lagrangian Methods for convex optimization and variational

inequalities 34 4.3. Dual and Primal-Dual Decomposition schemes 35 4.4. Primal Decomposition: Block Gauss-Seidel Schemes for Linearly

constrained Problems 39

vi

4.5. Convex Feasibility Problems 4.6. Bundle Methods in Nonsmooth Optimization

References

The Continuum Model of Transportation Problem Patrizia Daniele, Giovanna Idone and Antonino Maugeri

1. Introduction 2. Calculus of the solution References

The Economic Model for Demand-Supply Problems Patrizia Daniele and Antonino Maugeri

41 43 45

53

53 56 59

61

1. Introduction 61 2. The first phase: formalization of the equilibrium 62 3. The second phase: formalization of the equilibrium 67 4. The dependence of the second phase on the first one 70 5. General model 71 6. Example 72 References 77

Constrained Problems of Calculus of Variations Via Penal-ization Technique 79

Vladimir F. Demyanov

1. Introduction 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

References

79 80 81 83 86 88 88 94 97 99

100 100 103 106

vii

Variational Problems with Constraints Involving Higher-Order Derivatives 109

Vladimir F. Demyanov and Franco Giannessi

1. Introduction 109 2. Statement of the problem 110 3. An equivalent statement of the problem 111 4. Local minima 114 5. Properties of the function 'P 115

5.1. A classical variation of z 115 5.2. The case z if- Z 117 5.3. The case z E Z 120

6. Exact penalty functions 123 6.1. Properties of the function G 123 6.2. An Exact Penalty function 127

7. Necessary conditions for an Extremum 127 References 133

On the strong solvability of a unilateral boundary value problem for Nonlinear Parabolic Operators in the Plane 135

Rosalba Di Vincenzo

1. Introduction 135 2. Hypotheses and results 136 3. Preliminary results 137 4. Proof of the theorems 138 References 140

Solving a Special Class of Discrete Optimal Control Prob-lems Via a Parallel Interior-Point Method 141

Carla Dumzzi, Valeria Ruggiero and Gaetano Zanghimti

1. Introduction 2. Framework of the Method 3. Global convergence 4. A special class of discrete optimal control problems 5. Numerical experiments 6. Conclusions References

142 143 149 152 157 160 160

viii

Solving Large Scale Fixed Charge Network Flow Problems 163

Burak Ek§iofilu, Sandra Duni Ek§iofilu and Panos M. Pardalos

1. Introduction 164 2. Problem Definition and Formulation 166 3. Solution Procedure 167

3.1. The DSSP 167 3.2. Local Search 169

4. Computational Results 171 5. Concluding Remarks 181 References 181

Variable Projection Methods for Large-Scale Quadratic Optimization in data Analysis Applications 185

Emanuele Galligani, Valeria Ruggiero and Luca Zanni

1. Introduction 185 2. Large QP Problems in Training Support Vector Machines 188 3. Numerical Solution of Image Restoration Problem 193 4. A Bivariate Interpolation Problem 200 5. Conclusions 206 References 207

Strong solvability of boundary value problems in elasticity with Unilateral Constraints 213

Sofia Giuffre

1. Introduction 213 2. Basic assumptions and main results 215 3. Preliminary results 217 4. Proof of the theorems 218 References 223

Time Dependent Variational Inequalities - Some Recent Trends 225

Joachim Gwinner

1. Introduction 2. Time - an additional parameter in variational inequalities

226 229

ix

2.1. Time-dependent variational inequalities and quasi-variational in-equalities 230

2.2. Some classic results on the differentiability of the projection on closed convex subsets in Hilbert space 236

2.3. Time-dependent variational inequalities with memory terms 237 3. Ordinary Differential Inclusions with Convex Constraints: Sweeping Processes 240

3.1. Moving convex sets and systems with hysteresis 241 3.2. Sweeping processes and generalizations 242

4. Projected dynamical systems 247 4.1. Differentiability of the projection onto closed convex subsets

revisited 247 4.2. Projected dynamical systems and stationarity 250 4.3. Well-posedness for projected dynamical systems 251

5. Some Asymptotic Results 252 5.1. Some classical results 252 5.2. An asymptotic result for full discretization 253 5.3. Some convergence results for continuous-time subgradient proce-

dures for convex optimization 256 References 259

On the Contractibility of the Efficient and Weakly Efficient Sets in R2 265

Nguyen Quang Huy, Ta Duy Phuong and Nguyen Dong Yen

1. Introduction 265 2. Preliminaries 266 3. Topological structure of the efficient sets of compact convex sets 267 4. Example 276 References 278

Existence Theorems for a Class of Variational Inequalities and Applications to a Continuous Model of Transportation 281

Francesco Marino

1. Introduction 2. Continuous transportation model 3. Existence Theorem References

281 282 284 287

x

On Auxiliary Principle for Equilibrium Problems Giandomenico Mastroeni

289

1. Introduction 289 2. The auxiliary equilibrium problem 291 3. The auxiliary problem principle 293 4. Applications to variational inequalities and optimization problems 295 5. Concluding remarks 297 References 297

Multicriteria Spatial Price Networks: Statics and Dynam­ics

Anna Nagurney, June Dong and Ding Zhang

1. Introduction 2. The Multicriteria Spatial Price Model 3. Qualitative Properties 4. The Dynamics 5. The Discrete-Time Algorithm 6. Numerical Examples 7. Summary and Conclusions References

Non regular data in unilateral variational problems Pirro Oppezzi

299

299 301 306 309 311 314 318 319

323

1. Introduction 323 2. The approach by truncation and approximation 324 3. Renormalized formulation 328 4. Multivalued operators and more general measures 328 5. Uniqueness and convergence 330 References 331

Equilibrium Concepts in Transportation Networks: Generalized Wardrop Conditions and Variational For-mulations 333

Massimo Pappalardo and Mauro Passacantando

1. Introduction 2. Equilibrium model in a traffic network

333 334

References

Variational Geometry and Equilibrium Michael Patriksson and R. Tyrrell Rockafellar

xi

344

347

1. Introduction 347 2. Variational Inequalities and Normals to Convex Sets 349 3. Quasi-Variational Inequalities and Normals to General Sets 352 4. Calculus and Solution Perturbations 357 5. Application to an Equilibrium Model with Aggregation 361 References 367

On the Calculation of Equilibrium in Time Dependent Traffic Networks 369

Fabio Raciti

1. Introduction 2. Calculation of Equilibria 3. The algorithm 4. Applications and Examples 5. Conclusions References

Mechanical Equilibrium and Equilibrium Systems Tamas Rapcsak

369 370 371 372 376 376

379

1. Introduction 379 2. Physical motivation 380 3. Statement of the mechanical force equilibrium problem 381 4. The principle of virtual work 382 5. Characterization of the constraints 383 6. Quasi-variational inequalities (QVI) 384 7. Principle of virtual work in force fields under scleronomic and holo-nomic constraints 385 8. Dual form of the principle of virtual work in force field under sclero-nomic and holonomic constraints 388 9. Procedure for solving mechanical equilibrium problems 391 10. Existence of solutions 395 References 397

xii

False Numerical Convergence in Some Generalized Newton Methods 401

Stephen M. Robinson

1. Introduction 2. Some generalized Newton methods 3. False numerical convergence 4. An example 5. Avoiding false numerical convergence References

Distance to the Solution Set of an Inequality with an Increasing Function

Alex M. Rubinov

401 402 405 408 411 415

417

1. Introduction 417 2. Preliminaries 418 3. Distance to the solution set of the inequality with an arbitrary increas-ing function 420 4. Distance to the solution set of the inequality with an ICAR function 423 5. Inequalities with an increasing function defined on the entire space 427 6. Inequalities with a topical function 429 References 430

Transportation Networks with Capacity Constraints 433

Laum Scrimali

1. Introduction 2. Wardrop's generalized equilibrium condition 3. A triangular network 4. More about generalized equilibrium principle 5. Capacity constraints and paradox References

433 434 436 438 442 443

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 2 2000, 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