Optimal quadratic programming algorithms: with applications to variational inequalities
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Edition: | [1. ed.] |
| Series: | Springer optimization and its applications
23 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVII, 284 S. Ill., graph. Darst. |
| ISBN: | 9780387848051 9780387848068 |
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Record in the Search Index
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| adam_text |
OPTIMAL QUADRATIC PROGRAMMING ALGORITHMS WITH APPLICATIONS TO
VARIATIONAL INEQUALITIES BY ZDENEK DOSTAL VSB - TECHNICAL UNIVERSITY OF
OSTRAVA, CZECH REPUBLIC |)SPRI RINGER CONTENTS PREFACE VII PART I
BACKGROUND 1 LINEAR ALGEBRA 3 1.1 VECTORS 3 1.2 MATRICES AND MATRIX
OPERATIONS 5 1.3 MATRICES AND MAPPINGS 6 1.4 INVERSE AND GENERALIZED
INVERSE MATRICES 8 1.5 DIRECT METHODS FOR SOLVING LINEAR EQUATIONS 9 1.6
NORMS 12 1.7 SCALAR PRODUCTS 14 1.8 EIGENVALUES AND EIGENVECTORS 17 1.9
MATRIX DECOMPOSITIONS 19 1.10 PENALIZED MATRICES 22 2 OPTIMIZATION 27
2.1 OPTIMIZATION PROBLEMS AND SOLUTIONS 27 2.2 UNCONSTRAINED QUADRATIC
PROGRAMMING 28 2.2.1 QUADRATIC COST FUNCTIONS 28 2.2.2 UNCONSTRAINED
MINIMIZATION OF QUADRATIC FUNCTIONS . 29 2.3 CONVEXITY 31 2.3.1 CONVEX
QUADRATIC FUNCTIONS 32 2.3.2 LOCAL AND GLOBAL MINIMIZERS OF CONVEX
FUNCTION 34 2.3.3 EXISTENCE OF MINIMIZERS 35 2.3.4 PROJECTIONS TO CONVEX
SETS 36 2.4 EQUALITY CONSTRAINED PROBLEMS 38 2.4.1 OPTIMALITY CONDITIONS
39 2.4.2 EXISTENCE AND UNIQUENESS 41 2.4.3 KKT SYSTEMS 42 XIV CONTENTS
2.4.4 MIN-MAX, DUAL, AND SADDLE POINT PROBLEMS 44 2.4.5 SENSITIVITY 46
2.4.6 ERROR ANALYSIS 47 2.5 INEQUALITY CONSTRAINED PROBLEMS 49 2.5.1
POLYHEDRAL SETS 49 2.5.2 FARKAS'S LEMMA 50 2.5.3 NECESSARY OPTIMALITY
CONDITIONS FOR LOCAL SOLUTIONS . 51 2.5.4 EXISTENCE AND UNIQUENESS 52
2.5.5 OPTIMALITY CONDITIONS FOR CONVEX PROBLEMS 54 2.5.6 OPTIMALITY
CONDITIONS FOR BOUND CONSTRAINED PROBLEMS 55 2.5.7 MIN-MAX, DUAL, AND
SADDLE POINT PROBLEMS 55 2.6 EQUALITY AND INEQUALITY CONSTRAINED
PROBLEMS 57 2.6.1 OPTIMALITY CONDITIONS 58 2.6.2 EXISTENCE AND
UNIQUENESS 59 2.6.3 PARTIALLY BOUND AND EQUALITY CONSTRAINED PROBLEMS
. 59 2.6.4 DUALITY FOR DEPENDENT CONSTRAINTS 61 2.6.5 DUALITY FOR
SEMICOERCIVE PROBLEMS 64 2.7 LINEAR PROGRAMMING 69 2.7.1 SOLVABILITY AND
LOCALIZATION OF SOLUTIONS 69 2.7.2 DUALITY IN LINEAR PROGRAMMING 70 PART
II ALGORITHMS 3 CONJUGATE GRADIENTS FOR UNCONSTRAINED MINIMIZATION 73
3.1 CONJUGATE DIRECTIONS AND MINIMIZATION 74 3.2 GENERATING CONJUGATE
DIRECTIONS AND KRYLOV SPACES 77 3.3 CONJUGATE GRADIENT METHOD 78 3.4
RESTARTED CG AND THE GRADIENT METHOD 81 3.5 RATE OF CONVERGENCE AND
OPTIMALITY 82 3.5.1 MIN-MAX ESTIMATE 82 3.5.2 ESTIMATE IN THE CONDITION
NUMBER 84 3.5.3 CONVERGENCE RATE OF THE GRADIENT METHOD 86 3.5.4
OPTIMALITY 87 3.6 PRECONDITIONED CONJUGATE GRADIENTS 87 3.7
PRECONDITIONING BY CONJUGATE PROJECTOR 90 3.7.1 CONJUGATE PROJECTORS 90
3.7.2 MINIMIZATION IN SUBSPACE 91 3.7.3 CONJUGATE GRADIENTS IN CONJUGATE
COMPLEMENT 92 3.7.4 PRECONDITIONING EFFECT 94 3.8 CONJUGATE GRADIENTS
FOR MORE GENERAL PROBLEMS 96 3.9 CONVERGENCE IN PRESENCE OF ROUNDING
ERRORS 97 3.10 NUMERICAL EXPERIMENTS 98 3.10.1 BASIC CG AND
PRECONDITIONING 98 3.10.2 NUMERICAL DEMONSTRATION OF OPTIMALITY 99
CONTENTS XV 3.11 COMMENTS AND CONCLUSIONS 100 4 EQUALITY CONSTRAINED
MINIMIZATION 103 4.1 REVIEW OF ALTERNATIVE METHODS 105 4.2 PENALTY
METHOD 107 4.2.1 MINIMIZATION OF AUGMENTED LAGRANGIAN 108 4.2.2 AN
OPTIMAL FEASIBILITY ERROR ESTIMATE FOR HOMOGENEOUS CONSTRAINTS 109 4.2.3
APPROXIMATION ERROR AND CONVERGENCE ILL 4.2.4 IMPROVED FEASIBILITY ERROR
ESTIMATE 112 4.2.5 IMPROVED APPROXIMATION ERROR ESTIMATE 113 4.2.6
PRECONDITIONING PRESERVING GAP IN THE SPECTRUM 115 4.3 EXACT AUGMENTED
LAGRANGIAN METHOD 116 4.3.1 ALGORITHM 117 4.3.2 CONVERGENCE OF LAGRANGE
MULTIPLIERS 119 4.3.3 EFFECT OF THE STEPLENGTH 120 4.3.4 CONVERGENCE OF
THE FEASIBILITY ERROR 124 4.3.5 CONVERGENCE OF PRIMAL VARIABLES 124
4.3.6 IMPLEMENTATION 125 4.4 ASYMPTOTICALLY EXACT AUGMENTED LAGRANGIAN
METHOD 126 4.4.1 ALGORITHM 126 4.4.2 AUXILIARY ESTIMATES 127 4.4.3
CONVERGENCE ANALYSIS 128 4.5 ADAPTIVE AUGMENTED LAGRANGIAN METHOD 130
4.5.1 ALGORITHM 131 4.5.2 CONVERGENCE OF LAGRANGE MULTIPLIERS FOR LARGE
G 132 4.5.3 R-LINEAR CONVERGENCE FOR ANY INITIALIZATION OF G 134 4.6
SEMIMONOTONIC AUGMENTED LAGRANGIANS (SMALE) 135 4.6.1 SMALE ALGORITHM
136 4.6.2 RELATIONS FOR AUGMENTED LAGRANGIANS 137 4.6.3 CONVERGENCE AND
MONOTONICITY 139 4.6.4 LINEAR CONVERGENCE FOR LARGE GO 142 4.6.5
OPTIMALITY OF THE OUTER LOOP 143 4.6.6 OPTIMALITY OF SMALE WITH
CONJUGATE GRADIENTS 145 4.6.7 SOLUTION OF MORE GENERAL PROBLEMS 147 4.7
IMPLEMENTATION OF INEXACT AUGMENTED LAGRANGIANS 148 4.7.1 STOPPING,
MODIFICATION OF CONSTRAINTS, AND PRECONDITIONING 148 4.7.2
INITIALIZATION OF CONSTANTS 148 4.8 NUMERICAL EXPERIMENTS 150 4.8.1
UZAWA, EXACT AUGMENTED LAGRANGIANS, AND SMALE . 150 4.8.2 NUMERICAL
DEMONSTRATION OF OPTIMALITY 151 4.9 COMMENTS AND REFERENCES 152 XVI
CONTENTS 5 BOUND CONSTRAINED MINIMIZATION 155 5.1 REVIEW OF ALTERNATIVE
METHODS 157 5.2 KKT CONDITIONS AND RELATED INEQUALITIES 158 5.3 THE
WORKING SET METHOD WITH EXACT SOLUTIONS 160 5.3.1 AUXILIARY PROBLEMS 160
5.3.2 ALGORITHM 161 5.3.3 FINITE TERMINATION 164 5.4 POLYAK'S ALGORITHM
165 5.4.1 BASIC ALGORITHM 165 5.4.2 FINITE TERMINATION 166 5.4.3
CHARACTERISTICS OF POLYAK'S ALGORITHM 167 5.5 INEXACT POLYAK'S ALGORITHM
167 5.5.1 LOOKING AHEAD AND ESTIMATE 167 5.5.2 LOOKING AHEAD POLYAK'S
ALGORITHM 170 5.5.3 EASY RE-RELEASE POLYAK'S ALGORITHM 171 5.5.4
PROPERTIES OF MODIFIED POLYAK'S ALGORITHMS 172 5.6 GRADIENT PROJECTION
METHOD 173 5.6.1 CONJUGATE GRADIENT VERSUS GRADIENT PROJECTIONS 174
5.6.2 CONTRACTION IN THE EUCLIDEAN NORM 175 5.6.3 THE FIXED STEPLENGTH
GRADIENT PROJECTION METHOD . 177 5.6.4 QUADRATIC FUNCTIONS WITH
IDENTITY HESSIAN 178 5.6.5 DOMINATING FUNCTION AND DECREASE OF THE COST
FUNCTION 181 5.7 MODIFIED PROPORTIONING WITH GRADIENT PROJECTIONS 184
5.7.1 MPGP SCHEMA 184 5.7.2 RATE OF CONVERGENCE 186 5.8 MODIFIED
PROPORTIONING WITH REDUCED GRADIENT PROJECTIONS . 189 5.8.1 MPRGP
SCHEMA 189 5.8.2 RATE OF CONVERGENCE 190 5.8.3 RATE OF CONVERGENCE OF
PROJECTED GRADIENT 193 5.8.4 OPTIMALITY 197 5.8.5 IDENTIFICATION LEMMA
AND FINITE TERMINATION 198 5.8.6 FINITE TERMINATION FOR DUAL DEGENERATE
SOLUTION 201 5.9 IMPLEMENTATION OF MPRGP WITH OPTIONAL MODIFICATIONS 204
5.9.1 EXPANSION STEP WITH FEASIBLE HALF-STEP 204 5.9.2 MPRGP ALGORITHM
205 5.9.3 UNFEASIBLE MPRGP 206 5.9.4 CHOICE OF PARAMETERS 208 5.9.5
DYNAMIC RELEASE COEFFICIENT 209 5.10 PRECONDITIONING 210 5.10.1
PRECONDITIONING IN FACE 210 5.10.2 PRECONDITIONING BY CONJUGATE
PROJECTOR 212 5.11 NUMERICAL EXPERIMENTS 216 5.11.1 POLYAK, MPRGP, AND
PRECONDITIONED MPRGP 216 5.11.2 NUMERICAL DEMONSTRATION OF OPTIMALITY
217 5.12 COMMENTS AND REFERENCES 218 CONTENTS XVN 6 BOUND AND EQUALITY
CONSTRAINED MINIMIZATION 221 6.1 REVIEW OF THE METHODS FOR BOUND AND
EQUALITY CONSTRAINED PROBLEMS 222 6.2 SMALBE ALGORITHM FOR BOUND AND
EQUALITY CONSTRAINTS 223 6.2.1 KKT CONDITIONS AND PROJECTED GRADIENT 223
6.2.2 SMALBE ALGORITHM 223 6.3 INEQUALITIES INVOLVING THE AUGMENTED
LAGRANGIAN 225 6.4 MONOTONICITY AND FEASIBILITY 227 6.5 BOUNDEDNESS 229
6.6 CONVERGENCE 233 6.7 OPTIMALITY OF THE OUTER LOOP 235 6.8 OPTIMALITY
OF THE INNER LOOP 237 6.9 SOLUTION OF MORE GENERAL PROBLEMS 239 6.10
IMPLEMENTATION 240 6.11 SMALBE-M 241 6.12 NUMERICAL EXPERIMENTS 242
6.12.1 BALANCED REDUCTION OF FEASIBILITY AND GRADIENT ERRORS . 242
6.12.2 NUMERICAL DEMONSTRATION OF OPTIMALITY 243 6.13 COMMENTS AND
REFERENCES 244 PART III APPLICATIONS TO VARIATIONAL INEQUALITIES 7
SOLUTION OF A COERCIVE VARIATIONAL INEQUALITY BY FETI-DP METHOD 249 7.1
MODEL COERCIVE VARIATIONAL INEQUALITY 250 7.2 FETI-DP DOMAIN
DECOMPOSITION AND DISCRETIZATION 251 7.3 OPTIMALITY 254 7.4 NUMERICAL
EXPERIMENTS 255 7.5 COMMENTS AND REFERENCES 256 8 SOLUTION OF A
SEMICOERCIVE VARIATIONAL INEQUALITY BY TFETI METHOD 259 8.1 MODEL
SEMICOERCIVE VARIATIONAL INEQUALITY 260 8.2 TFETI DOMAIN DECOMPOSITION
AND DISCRETIZATION 261 8.3 NATURAL COARSE GRID 264 8.4 OPTIMALITY 265
8.5 NUMERICAL EXPERIMENTS 267 8.6 COMMENTS AND REFERENCES 269 REFERENCES
271 INDEX 281 |
| any_adam_object | 1 |
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| discipline | Mathematik |
| edition | [1. ed.] |
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| language | English |
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| series2 | Springer optimization and its applications |
| spelling | Dostál, Zdeněk Verfasser aut Optimal quadratic programming algorithms with applications to variational inequalities by Zdeněk Dostál [1. ed.] New York, NY Springer 2009 XVII, 284 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer optimization and its applications 23 Quadratic programming Quadratische Optimierung (DE-588)4130555-3 gnd rswk-swf Operations Research (DE-588)4043586-6 gnd rswk-swf Quadratische Optimierung (DE-588)4130555-3 s Operations Research (DE-588)4043586-6 s DE-604 Springer optimization and its applications 23 (DE-604)BV021746093 23 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3168939&prov=M&dok_var=1&dok_ext=htm Inhaltstext GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627591&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dostál, Zdeněk Optimal quadratic programming algorithms with applications to variational inequalities Springer optimization and its applications Quadratic programming Quadratische Optimierung (DE-588)4130555-3 gnd Operations Research (DE-588)4043586-6 gnd |
| subject_GND | (DE-588)4130555-3 (DE-588)4043586-6 |
| title | Optimal quadratic programming algorithms with applications to variational inequalities |
| title_auth | Optimal quadratic programming algorithms with applications to variational inequalities |
| title_exact_search | Optimal quadratic programming algorithms with applications to variational inequalities |
| title_full | Optimal quadratic programming algorithms with applications to variational inequalities by Zdeněk Dostál |
| title_fullStr | Optimal quadratic programming algorithms with applications to variational inequalities by Zdeněk Dostál |
| title_full_unstemmed | Optimal quadratic programming algorithms with applications to variational inequalities by Zdeněk Dostál |
| title_short | Optimal quadratic programming algorithms |
| title_sort | optimal quadratic programming algorithms with applications to variational inequalities |
| title_sub | with applications to variational inequalities |
| topic | Quadratic programming Quadratische Optimierung (DE-588)4130555-3 gnd Operations Research (DE-588)4043586-6 gnd |
| topic_facet | Quadratic programming Quadratische Optimierung Operations Research |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3168939&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018627591&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV021746093 |
| work_keys_str_mv | AT dostalzdenek optimalquadraticprogrammingalgorithmswithapplicationstovariationalinequalities |