Nonlinear programming and variational inequality problems: a unified approach
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1999
|
| Series: | Applied optimization
23 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 334 S. graph. Darst. |
| ISBN: | 0792354559 |
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| adam_text | IMAGE 1
NONLINEAR PROGRAMMING
AND VARIATIONAL INEQUALITY PROBLEMS A UNIFIED APPROACH
BY
MICHAEL PATRIKSSON
CHALMERS UNIVERSITY OF TECHNOLOGY, GOTHENBURG, SWEDEN
II
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
IMAGE 2
CONTENTS
PREFACE XI
NOTATION XIII
1 INTRODUCTION 1
1.1 THE VARIATIONAL INEQUALITY PROBLEM 1
1.1.1 INSTANCES OF THE PROBLEM 2
1.1.2 EXAMPLE APPLICATIONS 8
1.2 THE COST APPROXIMATION ALGORITHM 13
1.2.1 THE SUBPROBLEM PHASE 13
1.2.2 THE UPDATING PHASE 20
1.2.3 DISCUSSION 22
1.3 SCOPE AND PREVIEW 26
1.3.1 PREVIEW 26
1.3.2 SCOPE 36
2 TECHNICAL PRELIMINARIES 39
2.1 SOLUTIONS TO THE VARIATIONAL INEQUALITY PROBLEM 39
2.2 SOLUTIONS TO THE CA SUBPROBLEM 40
2.3 A POSTERIORI ERROR BOUNDS AND LOWER BOUNDS 42
2.4 DESCENT PROPERTIES 44
2.5 STEP LENGTH RULES 49
3 INSTANCES OF THE COST APPROXIMATION ALGORITHM 57
3.1 CLASSIC ALGORITHMS 57
3.1.1 LINEARIZATION METHODS 57
3.1.2 INTERIOR POINT ALGORITHMS 59
3.1.3 JACOBI AND GAUSS-SEIDEL METHODS 60
3.2 REGULARIZATION AND PROXIMAL POINT METHODS 61
3.2.1 REGULARIZATION METHODS 61
3.2.2 SPLITTING METHODS 71
3.3 DECOMPOSITION-COORDINATION METHODS 77
3.3.1 A PRIMAL ALGORITHM 78
3.3.2 A PRIMAL-DUAL ALGORITHM 79
IMAGE 3
VIII NONLINEAR PROGRAMMING AND VARIATIONAL INEQUALITY PROBLEMS
3.3.3 AN AUGMENTED LAGRANGEAN METHOD 80
3.4 DECOMPOSITION OF OPTIMIZATION PROBLEMS 81
3.5 RELATIONSHIPS AMONG ALGORITHM FRAMEWORKS 83
3.6 CA ALGORITHMS INVOLVING U 87
3.6.1 SUBGRADIENT ALGORITHMS 87
3.6.2 PERTURBED CA ALGORITHMS 88
3.7 CONTINUOUS CA ALGORITHMS 90
3.8 A FINAL REMARK 92
4 MERIT FUNCTIONS FOR VARIATIONAL INEQUALITY PROBLEMS 95
4.1 INTRODUCTION 95
4.2 A CLASS OF MERIT FUNCTIONS FOR VARIATIONAL INEQUALITIES 96
4.3 PROPERTIES OF THE MERIT FUNCTION IP 99
4.4 INSTANCES OF THE MERIT FUNCTION IP 104
4.4.1 THE PRIMAL AND DUAL GAP FUNCTIONS 104
4.4.2 SOME DIFFERENTIABLE MERIT FUNCTIONS 106
4.4.3 UNCONSTRAINED AND COMPLEMENTARITY FORMULATIONS 107
4.4.4 MERIT FUNCTIONS FOR VARIATIONAL INEQUALITY PROBLEMS WITH
MULTI-VALUED OPERATORS 110
4.5 STATIONARITY CONDITIONS AND DESCENT PROPERTIES 112
4.5.1 COST APPROXIMATING MAPPINGS INDEPENDENT OF A: 112
4.5.2 COST APPROXIMATING MAPPINGS PARAMETERIZED BY X . . .. 115 4.5.3
COMBINED COST APPROXIMATING MAPPINGS 117
4.5.4 DESCENT FROM TRUNCATED CA SUBPROBLEMS 120
4.6 PRIMAL-DUAL VARIATIONAL INEQUALITIES 122
4.6.1 INTRODUCTION 122
4.6.2 A PRIMAL MERIT FUNCTION 126
4.6.3 COMBINED COST APPROXIMATING MAPPINGS 130
5 CONVERGENCE OF THE CA ALGORITHM FOR NONLINEAR PROGRAMS 135
5.1 CONVERGENCE UNDER AN EXACT LINE SEARCH 135
5.2 CONVERGENCE UNDER THE ARMIJO AND MODIFIED ARMIJO RULE 138
5.2.1 APPLICATION TO DIFFERENTIABLE OPTIMIZATION 138
5.2.2 APPLICATION TO NON-DIFFERENTIABLE OPTIMIZATION 140
5.3 A TRUNCATION SCHEME FOR THE CA SUBPROBLEM 141
5.3.1 THE TRUNCATION SCHEME 141
5.3.2 CONVERGENCE UNDER AN EXACT LINE SEARCH 143
5.3.3 CONVERGENCE UNDER THE ARMIJO RULE 145
5.4 CONVERGENCE UNDER VARIOUS STEP LENGTH RULES 146
5.4.1 EXACT SUBPROBLEM SOLUTIONS 146
5.4.2 INEXACT SUBPROBLEM SOLUTIONS 151
5.4.3 A PERTURBED CA ALGORITHM 151
5.4.4 APPLICATION TO CONVEX PROBLEMS WITH EXPLICIT CONSTRAINTS . 152 5.5
A NON-MONOTONE CA ALGORITHM 154
5.5.1 INTRODUCTION 154
5.5.2 THE NON-MONOTONE CA ALGORITHM . . . * 155
IMAGE 4
CONTENTS IX
5.5.3 CONVERGENCE OF THE NON-MONOTONE CA ALGORITHM 157
5.6 LINEAR CONVERGENCE RESULTS 161
5.6.1 APPLICATION TO DIFFERENTIABLE OPTIMIZATION 161
5.6.2 APPLICATION TO NON-DIFFERENTIABLE OPTIMIZATION 165
6 CONVERGENCE OF THE CA ALGORITHM FOR VARIATIONAL INEQUALITY PROBLEMS
169
6.1 ITERATION-INDEPENDENT COST APPROXIMATION 169
6.1.1 CONVERGENCE UNDER AN EXACT LINE SEARCH 169
6.1.2 CONVERGENCE UNDER DIFFERENT STEP LENGTH RULES 170
6.1.3 LINEAR CONVERGENCE 172
6.2 ITERATION-DEPENDENT COST APPROXIMATION 173
6.2.1 PRIMAL APPLICATION 173
6.2.2 PRIMAL-DUAL APPLICATION 176
6.3 NON-DESCENT CA METHODS 182
6.3.1 ITERATION-INDEPENDENT COST APPROXIMATION 183
6.3.2 ITERATION-DEPENDENT COST APPROXIMATION 185
6.4 AVERAGING SCHEMES AND ERGODIC SEQUENCES 186
6.4.1 NON-EXPANSIVENESS OF THE CA SUBPROBLEM MAPPING . . .. 187 6.4.2
ITERATIVE AND ERGODIC SCHEMES 188
7 FINITE IDENTIFICATION OF ACTIVE CONSTRAINTS AND OF SOLUTIONS 191
7.1 FINITE IDENTIFICATION OF ACTIVE CONSTRAINTS 191
7.1.1 FACIAL GEOMETRY 192
7.1.2 THE PROJECTED GRADIENT AND STATIONARITY CONDITIONS . . .. 193
7.1.3 NON-DEGENERACY 194
7.1.4 IDENTIFICATION RESULTS 196
7.2 FINITE IDENTIFICATION OF SOLUTIONS 203
7.2.1 SHARP SOLUTIONS 203
7.2.2 FINITE TERMINATION OF THE CA ALGORITHM 204
8 PARALLEL AND SEQUENTIAL DECOMPOSITION CA ALGORITHMS 211
8.1 INTRODUCTION 211
8.1.1 THE PROBLEM UNDER STUDY 212
8.1.2 ADAPTING CA TO THE CARTESIAN PRODUCT STRUCTURE 214
8.1.3 SCOPE AND PREVIEW 215
8.2 SEQUENTIAL DECOMPOSITION CA ALGORITHMS 217
8.3 SYNCHRONIZED PARALLEL CA ALGORITHMS 218
8.3.1 SYNCHRONIZED PARALLEL COMPUTATIONS - 219
8.3.2 THE SYNCHRONIZED PARALLEL ALGORITHM 220
8.4 PARTIALLY ASYNCHRONOUS PARALLEL DECOMPOSITION CA ALGORITHMS . . 223
8.4.1 ASYNCHRONOUS PARALLEL COMPUTATIONS 223
8.4.2 THE PARTIALLY ASYNCHRONOUS PARALLEL ALGORITHM 225
8.5 CONVERGENCE OF THE SEQUENTIAL DECOMPOSITION ALGORITHM 226
8.5.1 ON THE SEPARABILITY ASSUMPTION OF U 226
8.5.2 CONVERGENCE UNDER AN EXACT LINE SEARCH 228
IMAGE 5
X NONLINEAR PROGRAMMING AND VARIATIONAL INEQUALITY PROBLEMS
8.5.3 CONVERGENCE OF A TRUNCATED ALGORITHM 229
8.5.4 ESSENTIALLY CYCLIC DECOMPOSITION CA ALGORITHMS 231
8.5.5 LINEAR CONVERGENCE 236
8.6 CONVERGENCE OF THE SYNCHRONIZED PARALLEL ALGORITHMS 237
8.6.1 CONVERGENCE OF A TRUNCATED ALGORITHM 238
8.6.2 CONVERGENCE UNDER DIFFERENT STEP LENGTH RULES 238
8.7 CONVERGENCE OF THE PARTIALLY ASYNCHRONOUS ALGORITHM 239
8.7.1 CONVERGENCE RESULTS 239
8.7.2 QUALITATIVE ANALYSIS 243
8.8 VARIATIONAL INEQUALITY PROBLEMS OVER CARTESIAN PRODUCT SETS . . .
244
9 A COLUMN GENERATION/SIMPLICIAL DECOMPOSITION ALGORITHM 253
9.1 COLUMN GENERATION APPROACHES 253
9.1.1 BACKGROUND 253
9.1.2 INNER REPRESENTATION 254
9.1.3 SIMPLICIAL DECOMPOSITION 256
9.2 THE COLUMN GENERATION CA ALGORITHM 262
9.2.1 UPDATING THE INNER APPROXIMATION 262
9.2.2 THE COLUMN GENERATION CA ALGORITHM 263
9.2.3 INSTANCES 264
9.3 CONVERGENCE FOR NONLINEAR PROGRAMS 267
9.3.1 SET AUGMENTATION 267
9.3.2 CONVERGENCE UNDER EXACT SOLUTIONS OF RMP 270
9.3.3 CONVERGENCE OF A TRUNCATED ALGORITHM 272
9.3.4 AN ALGORITHM WITH GENERAL COLUMN DROPPING RULES 273
9.4 CONVERGENCE FOR VARIATIONAL INEQUALITY PROBLEMS 273
9.4.1 CONVERGENCE UNDER AN EXACT SOLUTION OF RMP 273
9.4.2 AN ALGORITHM WITH GENERAL COLUMN DROPPING RULES 274
9.4.3 A PRIMAL-DUAL APPLICATION 275
A DEFINITIONS 277
REFERENCES 283
INDEX 325
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:33:34Z |
| institution | BVB |
| isbn | 0792354559 |
| language | English |
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| series | Applied optimization |
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| spelling | Patriksson, Michael Verfasser aut Nonlinear programming and variational inequality problems a unified approach by Michael Patriksson Dordrecht [u.a.] Kluwer 1999 XIV, 334 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied optimization 23 Niet-lineaire programmering gtt Programação não linear larpcal Variatieongelijkheden gtt Nonlinear programming Variational inequalities (Mathematics) Nichtlineare Optimierung (DE-588)4128192-5 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Variationsungleichung (DE-588)4187420-1 gnd rswk-swf Variationsungleichung (DE-588)4187420-1 s Approximation (DE-588)4002498-2 s DE-604 Nichtlineare Optimierung (DE-588)4128192-5 s Applied optimization 23 (DE-604)BV010841718 23 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008693019&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Patriksson, Michael Nonlinear programming and variational inequality problems a unified approach Applied optimization Niet-lineaire programmering gtt Programação não linear larpcal Variatieongelijkheden gtt Nonlinear programming Variational inequalities (Mathematics) Nichtlineare Optimierung (DE-588)4128192-5 gnd Approximation (DE-588)4002498-2 gnd Variationsungleichung (DE-588)4187420-1 gnd |
| subject_GND | (DE-588)4128192-5 (DE-588)4002498-2 (DE-588)4187420-1 |
| title | Nonlinear programming and variational inequality problems a unified approach |
| title_auth | Nonlinear programming and variational inequality problems a unified approach |
| title_exact_search | Nonlinear programming and variational inequality problems a unified approach |
| title_full | Nonlinear programming and variational inequality problems a unified approach by Michael Patriksson |
| title_fullStr | Nonlinear programming and variational inequality problems a unified approach by Michael Patriksson |
| title_full_unstemmed | Nonlinear programming and variational inequality problems a unified approach by Michael Patriksson |
| title_short | Nonlinear programming and variational inequality problems |
| title_sort | nonlinear programming and variational inequality problems a unified approach |
| title_sub | a unified approach |
| topic | Niet-lineaire programmering gtt Programação não linear larpcal Variatieongelijkheden gtt Nonlinear programming Variational inequalities (Mathematics) Nichtlineare Optimierung (DE-588)4128192-5 gnd Approximation (DE-588)4002498-2 gnd Variationsungleichung (DE-588)4187420-1 gnd |
| topic_facet | Niet-lineaire programmering Programação não linear Variatieongelijkheden Nonlinear programming Variational inequalities (Mathematics) Nichtlineare Optimierung Approximation Variationsungleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008693019&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010841718 |
| work_keys_str_mv | AT patrikssonmichael nonlinearprogrammingandvariationalinequalityproblemsaunifiedapproach |