Nonsmooth equations in optimization: regularity, calculus, methods, and applications
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Dordrecht [u.a.]
Kluwer
2002
|
| Schriftenreihe: | Nonconvex optimization and its applications
60 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (p. 311-324) and index |
| Beschreibung: | XXVIII, 329 S. Ill. |
| ISBN: | 1402005504 |
Internformat
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| 245 | 1 | 0 | |a Nonsmooth equations in optimization |b regularity, calculus, methods, and applications |c by Diethard Klatte and Bernd Kummer |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 2002 | |
| 300 | |a XXVIII, 329 S. |b Ill. | ||
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| 490 | 1 | |a Nonconvex optimization and its applications |v 60 | |
| 500 | |a Includes bibliographical references (p. 311-324) and index | ||
| 650 | 4 | |a Optimisation non différentiable | |
| 650 | 4 | |a Nonsmooth optimization | |
| 650 | 0 | 7 | |a Nichtglatte Optimierung |0 (DE-588)4120798-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
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|---|---|
| adam_text | NONSMOOTH EQUATIONS IN OPTIMIZATION REGULARITY, CALCULUS, METHODS AND
APPLICATIONS BY DIETHARD KLATTE INSTITUTE FOR OPERATIONS RESEARCH AND
MATHEMATICAL METHODS OF ECONOMICS, UNIVERSITY OF ZURICH, SWITZERLAND AND
BEMD KUMMER INSTITUTE OF MATHEMATICS, FACULTY OF MATHEMATICS AND NATURAL
SCIENCES II, HUMBOLDT UNIVERSITY BERLIN, GERMANY KLUWER ACADEMIC
PUBLISHERS DORDRECHT / BOSTON / LONDON CONTENTS INTRODUCTION XI LIST OF
RESULTS XIX BASIC NOTATION XXV 1 BASIC CONCEPTS 1 1.1 FORMAL SETTINGS 1
1.2 MULTIFUNCTIONS AND DERIVATIVES 2 1.3 PARTICULAR LOCALLY LIPSCHITZ
FUNCTIONS AND RELATED DEFINITIONS . 4 GENERALIZED JACOBIANS OF LOCALLY
LIPSCHITZ FUNCTIONS 4 PSEUDO-SMOOTHNESS AND DF 4 PIECEWISE C 1
FUNCTIONS 5 NCP FUNCTIONS 5 1.4 DEFINITIONS OF REGULARITY 6 DEFINITIONS
OF LIPSCHITZ PROPERTIES 6 REGULARITY DEFINITIONS 7 FUNCTIONS AND
MULTIFUNCTIONS 9 1.5 RELATED DEFINITIONS 10 TYPES OF SEMICONTINUITY 10
METRIC, PSEUDO-, UPPER REGULARITY; OPENNESS WITH LINEAR RATE . 12
CALMNESS AND UPPER REGULARITY AT A SET 13 1.6 FIRST MOTIVATIONS 14
PARAMETRIC GLOBAL MINIMIZERS 15 PARAMETRIC LOCAL MINIMIZERS 16
EPI-CONVERGENCE 17 2 REGULARITY AND CONSEQUENCES 19 2.1 UPPER REGULARITY
AT POINTS AND SETS 19 CHARACTERIZATION BY INCREASING FUNCTIONS 19
OPTIMALITY CONDITIONS 25 LINEAR INEQUALITY SYSTEMS WITH VARIABLE MATRIX
28 APPLICATION TO LAGRANGE MULTIPLIERS 30 UPPER REGULARITY AND NEWTON S
METHOD 31 VI CONTENTS 2.2 PSEUDO-REGULARITY 32 2.2.1 THE FAMILY OF
INVERSE FUNCTIONS 34 2.2.2 EKELAND POINTS AND UNIFORM LOWER
SEMICONTINUITY . . . . 37 2.2.3 SPECIAL MULTIFUNCTIONS 43 LEVEL SETS
OFL.S.C. FUNCTIONS 43 CONE CONSTRAINTS 44 LIPSCHITZ OPERATORS WITH I M A
G E S IN HILBERT SPACES . . . . 46 NECESSARY OPTIMALITY CONDITIONS 47
2.2.4 INTERSECTION MAPS AND EXTENSION OF MFCQ 49 INTERSECTION WITH A
QUASI-LIPSCHITZ MULTIFUNCTION 49 SPECIAL CASES 54 INTERSECTIONS WITH
HYPERFACES 58 3 CHARACTERIZATIONS OF REGULARITY BY DERIVATIVES 61 3.1
STRONG REGULARITY AND THIBAULT S LIMIT SETS 61 3.2 UPPER REGULARITY AND
CONTINGENT DERIVATIVES 63 3.3 PSEUDO-REGULARITY AND GENERALIZED
DERIVATIVES 63 CONTINGENT DERIVATIVES 64 PROPER MAPPINGS 64 CLOSED
MAPPINGS 64 CODERIVATIVES 66 VERTICAL NORMALS 67 4 NONLINEAR VARIATIONS
AND IMPLICIT FUNCTIONS 71 4.1 SUCCESSIVE APPROXIMATION AND PERSISTENCE
OF PSEUDO-REGULARITY . 72 4.2 PERSISTENCE OF UPPER REGULARITY 77
PERSISTENCE BASED ON KAKUTANI S FIXED POINT THEOREM 77 PERSISTENCE BASED
ON GROWTH CONDITIONS 79 4.3 IMPLICIT FUNCTIONS 82 5 CLOSED MAPPINGS IN
FINITE DIMENSION 89 5.1 CLOSED MULTIFUNCTIONS IN FINITE DIMENSION 89
5.1.1 SUMMARY OF REGULARITY CONDITIONS VIA DERIVATIVES . . . . 89 5.1.2
REGULARITY OF THE CONVEX SUBDIFFERENTIAL 92 5.2 CONTINUOUS AND LOCALLY
LIPSCHITZ FUNCTIONS 93 5.2.1 PSEUDO-REGULARITY AND EXACT PENALIZATION 94
5.2.2 SPECIAL STATEMENTS FOR M = N 96 5.2.3 CONTINUOUS SELECTIONS OF
PSEUDO-LIPSCHITZ MAPS 99 5.3 IMPLICIT LIPSCHITZ FUNCTIONS ONR 100 6
ANALYSIS OF GENERALIZED DERIVATIVES 105 6.1 GENERAL PROPERTIES FOR
ABSTRACT AND POLYHEDRAL MAPPINGS . . . . 105 6.2 DERIVATIVES FOR
LIPSCHITZ FUNCTIONS IN FINITE DIMENSION 110 6.3 RELATIONS BETWEEN TF AND
DF 113 6.4 CHAIN RULES OF EQUATION TYPE 115 6.4.1 CHAIN RULES FOR TF AND
CF WITH / * C 0 1 115 CONTENTS VII 6.4.2 NEWTON MAPS AND
SEMISMOOTHNESS 121 6.5 MEAN VALUE THEOREMS, TAYLOR EXPANSION AND
QUADRATIC GROWTH 131 6.6 CONTINGENT DERIVATIVES OF IMPLICIT (MULTI-)
FUNCTIONS AND STA- TIONARY POINTS 136 6.6.1 CONTINGENT DERIVATIVE OF AN
IMPLICIT (MULTI-)FUNCTION . . 137 6.6.2 CONTINGENT DERIVATIVE OF A
GENERAL STATIONARY POINT MAP 141 7 CRITICAL POINTS AND GENERALIZED
KOJIMA*FUNCTIONS 149 7.1 MOTIVATION AND DEFINITION 149 KKT POINTS AND
CRITICAL POINTS IN KOJIMA S SENSE 150 GENERALIZED KOJIMA-FUNCTIONS -
DEFINITION 151 7.2 EXAMPLES AND CANONICAL PARAMETRIZATIONS 154 THE
SUBDIFFERENTIAL OF A CONVEX MAXIMUM FUNCTION 154 COMPLEMENTARITY
PROBLEMS 156 GENERALIZED EQUATIONS 157 NASH EQUILIBRIA 159 PIECEWISE
AFSNE BIJECTIONS 160 7.3 DERIVATIVES AND REGULARITY OF GENERALIZED
KOJIMA-FUNCTIONS . . 160 PROPERTIES OFN 160 FORMULAS FOR GENERALIZED
DERIVATIVES 164 REGULARITY CHARACTERIZATIONS BY STABILITY SYSTEMS 167
GEOMETRICAL INTERPRETATION 168 7.4 DISCUSSION OF PARTICULAR CASES 170
7.4.1 THE CASE OF SMOOTH DATA 170 7.4.2 STRONG REGULARITY OF
COMPLEMENTARITY PROBLEMS 175 7.4.3 REVERSED INEQUALITIES 177 7.5
PSEUDO-REGULARITY VERSUS STRONG REGULARITY 178 8 PARAMETRIC OPTIMIZATION
PROBLEMS 183 8.1 THE BASIC MODEL 185 8.2 CRITICAL POINTS UNDER
PERTURBATIONS 187 8.2.1 STRONG REGULARITY 187 GEOMETRICAL INTERPRETATION
189 DIRECT PERTURBATIONS FOR THE QUADRATIC APPROXIMATION . . 190 STRONG
REGULARITY OF LOCAL MINIMIZERS UNDER LICQ . . . . 191 8.2.2 LOCAL UPPER
LIPSCHITZ CONTINUITY 193 REFORMULATION OF THE C-STABILITY SYSTEM 194
GEOMETRICAL INTERPRETATION 196 DIRECT PERTURBATIONS FOR THE QUADRATIC
APPROXIMATION . . 197 8.3 STATIONARY AND OPTIMAL SOLUTIONS UNDER
PERTURBATIONS 198 8.3.1 CONTINGENT DERIVATIVE OF THE STATIONARY POINT
MAP . . . . 199 THE CASE OF LOCALLY LIPSCHITZIAN F 200 THE SMOOTH CASE
202 8.3.2 LOCAL UPPER LIPSCHITZ CONTINUITY 203 INJECTIVITY AND
SECOND-ORDER CONDITIONS 205 VIII CONTENTS CONDITIONS VIA QUADRATIC
APPROXIMATION 208 LINEARLY CONSTRAINED PROGRAMS 209 8.3.3 UPPER
REGULARITY 210 UPPER REGULARITY OF ISOLATED MINIMIZERS 211 SECOND-ORDER
OPTIMALITY CONDITIONS FOR C 1 1 PROGRAMS . 215 8.3.4 STRONGLY REGULAR
AND PSEUDO-LIPSCHITZ STATIONARY POINTS 217 STRONG REGULARITY . 217
PSEUDO-LIPSCHITZ PROPERTY 220 8.4 TAYLOR EXPANSION OF CRITICAL VALUES
221 8.4.1 MARGINAL MAP UNDER CANONICAL PERTURBATIONS 222 8.4.2 MARGINAL
MAP UNDER NONLINEAR PERTURBATIONS 225 FORMULAS UNDER UPPER REGULARITY OF
STATIONARY POINTS . . 225 FORMULAS UNDER STRONG REGULARITY 227 FORMULAS
IN TERMS OF THE CRITICAL VALUE FUNCTION GIVEN UNDER CANONICAL
PERTURBATIONS 229 9 DERIVATIVES AND REGULARITY OF FURTHER NONSMOOTH MAPS
231 9.1 GENERALIZED DERIVATIVES FOR POSITIVELY HOMOGENEOUS FUNCTIONS . .
231 9.2 NCP FUNCTIONS 236 CASE (I): DESCENT METHODS 237 CASE (II):
NEWTON METHODS 238 9.3 THE C-DERIVATIVE OF THE MAX-FUNCTION
SUBDIFFERENTIAL 241 CONTINGENT LIMITS 243 CHARACTERIZATION OF C D C F
FOR MAX-FUNCTIONS: SPECIAL STRUCTURE . 244 CHARACTERIZATION OFC8 C F FOR
MAX-FUNCTIONS: GENERAL STRUCTURE 251 APPLICATION 1 . 253 APPLICATION 2
254 10 NEWTON S METHOD FOR LIPSCHITZ EQUATIONS 257 10.1 LINEAR AUXILIARY
PROBLEMS 257 10.1.1 DENSE SUBSETS AND APPROXIMATIONS OF M 260 10.1.2
PARTICULAR SETTINGS 261 10.1.3 REALIZATIONS FOR LOCPC 1 AND NCP
FUNCTIONS 262 10.2 THE USUAL NEWTON METHOD FOR PC 1 FUNCTIONS 265 10.3
NONLINEAR AUXILIARY PROBLEMS 265 10.3.1 CONVERGENCE 267 10.3.2 NECESSITY
OF THE CONDITIONS 270 11 PARTICULAR NEWTON REALIZATIONS AND SOLUTION
METHODS 275 11.1 PERTURBED KOJIMA SYSTEMS 276 QUADRATIC PENALTIES 276
LOGARITHMIC BARRIERS 276 11.2 PARTICULAR NEWTON REALIZATIONS AND
SQP-MODELS 278 CONTENTS IX 12 BASIC EXAMPLES AND EXERCISES 287 12.1
BASIC EXAMPLES 287 12.2 EXERCISES 296 APPENDIX 303 EKELAND S VARIATIONAL
PRINCIPLE . . . 303 APPROXIMATION BY DIRECTIONAL DERIVATIVES 304 PROOF
OF TF = T(NM) = NTM + TNM 306 CONSTRAINT QUALIFICATIONS 307 BIBLIOGRAPHY
311 INDEX 325
|
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| id | DE-604.BV014515881 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:03:13Z |
| institution | BVB |
| isbn | 1402005504 |
| language | English |
| lccn | 2002071067 |
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| oclc_num | 49859894 |
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| owner | DE-20 DE-703 DE-634 DE-11 |
| owner_facet | DE-20 DE-703 DE-634 DE-11 |
| physical | XXVIII, 329 S. Ill. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Kluwer |
| record_format | marc |
| series | Nonconvex optimization and its applications |
| series2 | Nonconvex optimization and its applications |
| spelling | Klatte, Diethard Verfasser aut Nonsmooth equations in optimization regularity, calculus, methods, and applications by Diethard Klatte and Bernd Kummer Dordrecht [u.a.] Kluwer 2002 XXVIII, 329 S. Ill. txt rdacontent n rdamedia nc rdacarrier Nonconvex optimization and its applications 60 Includes bibliographical references (p. 311-324) and index Optimisation non différentiable Nonsmooth optimization Nichtglatte Optimierung (DE-588)4120798-1 gnd rswk-swf Nichtglatte Optimierung (DE-588)4120798-1 s DE-604 Kummer, Bernd Sonstige oth Nonconvex optimization and its applications 60 (DE-604)BV010085908 60 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009886432&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Klatte, Diethard Nonsmooth equations in optimization regularity, calculus, methods, and applications Nonconvex optimization and its applications Optimisation non différentiable Nonsmooth optimization Nichtglatte Optimierung (DE-588)4120798-1 gnd |
| subject_GND | (DE-588)4120798-1 |
| title | Nonsmooth equations in optimization regularity, calculus, methods, and applications |
| title_auth | Nonsmooth equations in optimization regularity, calculus, methods, and applications |
| title_exact_search | Nonsmooth equations in optimization regularity, calculus, methods, and applications |
| title_full | Nonsmooth equations in optimization regularity, calculus, methods, and applications by Diethard Klatte and Bernd Kummer |
| title_fullStr | Nonsmooth equations in optimization regularity, calculus, methods, and applications by Diethard Klatte and Bernd Kummer |
| title_full_unstemmed | Nonsmooth equations in optimization regularity, calculus, methods, and applications by Diethard Klatte and Bernd Kummer |
| title_short | Nonsmooth equations in optimization |
| title_sort | nonsmooth equations in optimization regularity calculus methods and applications |
| title_sub | regularity, calculus, methods, and applications |
| topic | Optimisation non différentiable Nonsmooth optimization Nichtglatte Optimierung (DE-588)4120798-1 gnd |
| topic_facet | Optimisation non différentiable Nonsmooth optimization Nichtglatte Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009886432&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010085908 |
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