Optimization in function spaces: with stability considerations in Orlicz spaces
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
De Gruyter
2011
|
| Series: | De Gruyter series in nonlinear analysis and applications
13 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIV, 388 S. Ill. - graph. Darst. 240 mm x 170 mm |
| ISBN: | 9783110250206 |
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MARC
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| 100 | 1 | |a Kosmol, Peter |d 1942- |e Verfasser |0 (DE-588)106833189 |4 aut | |
| 245 | 1 | 0 | |a Optimization in function spaces |b with stability considerations in Orlicz spaces |c Peter Kosmol ; Dieter Müller-Wichards |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2011 | |
| 300 | |a XIV, 388 S. |b Ill. - graph. Darst. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
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IMAGE 1
CONTENTS
PREFACE V
1 APPROXIMATION IN ORLICZ SPACES 1
1.1 INTRODUCTION 1
1.2 A BRIEF FRAMEWORK FOR APPROXIMATION IN ORLICZ SPACES 7
1.3 APPROXIMATION IN C{T) 9
1.3.1 CHEBYSHEV APPROXIMATIONS IN C(T) 10
1.3.2 APPROXIMATION IN HAAR SUBSPACES 12
1.4 DISCRETE ^-APPROXIMATIONS 20
1.4.1 DISCRETE CHEBYSHEV APPROXIMATION 20
1.4.2 LINEAR L*-APPROXIMATION FOR FINITE YOUNG FUNCTIONS 24 1.5
DETERMINATION OF THE LINEAR L*-APPROXIMATION 28
1.5.1 SYSTEM OF EQUATIONS FOR THE COEFFICIENTS 28
1.5.2 THE METHOD OF KARLOVITZ 29
2 POLYA ALGORITHMS IN ORLICZ SPACES 34
2.1 THE CLASSICAL POLYA ALGORITHM 34
2.2 GENERALIZED POLYA ALGORITHM 34
2.3 POLYA ALGORITHM FOR THE DISCRETE CHEBYSHEV APPROXIMATION 35 2.3.1
THE STRICT APPROXIMATION AS THE LIMIT OF POLYA ALGORITHMS . 37 2.3.2
ABOUT THE CHOICE OF THE YOUNG FUNCTIONS 39
2.3.3 NUMERICAL EXECUTION OF THE POLYA ALGORITHM 40
2.4 STABILITY OF POLYA ALGORITHMS IN ORLICZ SPACES 41
2.5 CONVERGENCE ESTIMATES AND ROBUSTNESS 45
2.5.1 TWO-STAGE OPTIMIZATION 47
2.5.2 CONVERGENCE ESTIMATES 49
2.6 A POLYA-REMEZ ALGORITHM IN C(T) 58
2.7 SEMI-INFINITE OPTIMIZATION PROBLEMS 64
2.7.1 SUCCESSIVE APPROXIMATION OF THE RESTRICTION SET 64
3 CONVEX SETS AND CONVEX FUNCTIONS 72
3.1 GEOMETRY OF CONVEX SETS 72
3.2 CONVEX FUNCTIONS 76
3.3 DIFFERENCE QUOTIENT AND DIRECTIONAL DERIVATIVE 80
3.3.1 GEOMETRY OF THE RIGHT-SIDED DIRECTIONAL DERIVATIVE 81 3.4
NECESSARY AND SUFFICIENT OPTIMALITY CONDITIONS 85
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1007280395
DIGITALISIERT DURCH
IMAGE 2
XUE CONTENTS
3.4.1 NECESSARY OPTIMALITY CONDITIONS 85
3.4.2 SUFFICIENT CONDITION: CHARACTERIZATION THEOREM OF CONVEX
OPTIMIZATION 85
3.5 CONTINUITY OF CONVEX FUNCTIONS 86
3.6 FRE'CHET DIFFERENTIABILITY 87
3.7 CONVEX FUNCTIONS IN K" 88
3.8 CONTINUITY OF THE DERIVATIVE 91
3.9 SEPARATION THEOREMS 95
3.10 SUBGRADIENTS 99
3.11 CONJUGATE FUNCTIONS 102
3.12 THEOREM OF FENCHEL 105
3.13 EXISTENCE OF MINIMAL SOLUTIONS FOR CONVEX OPTIMIZATION 109 3.14
LAGRANGE MULTIPLIERS I LL
4 NUMERICAL TREATMENT OF NON-LINEAR EQUATIONS AND OPTIMIZATION PROBLEMS
115
4.1 NEWTON METHOD 116
4.2 SECANT METHODS 118
4.3 GLOBAL CONVERGENCE 120
4.3.1 DAMPED NEWTON METHOD 122
4.3.2 GLOBALIZATION OF SECANT METHODS FOR EQUATIONS 124
4.3.3 SECANT METHOD FOR MINIMIZATION 126
4.4 A MATRIX-FREE NEWTON METHOD 128
5 STABILITY AND TWO-STAGE OPTIMIZATION PROBLEMS 129
5.1 LOWER SEMI-CONTINUOUS CONVERGENCE AND STABILITY 129
5.1.1 LOWER SEMI-EQUICONTINUITY AND LOWER SEMI-CONTINUOUS CONVERGENCE
130
5.1.2 LOWER SEMI-CONTINUOUS CONVERGENCE AND CONVERGENCE OFEPIGRAPHS 131
5.2 STABILITY FOR MONOTONE CONVERGENCE 132
5.3 CONTINUOUS CONVERGENCE AND STABILITY FOR CONVEX FUNCTIONS 134 5.3.1
STABILITY THEOREMS 141
5.4 CONVEX OPERATORS 147
5.5 QUANTITATIVE STABILITY CONSIDERATIONS IN R" 153
5.6 TWO-STAGE OPTIMIZATION 156
5.6.1 SECOND STAGES AND STABILITY FOR EPSILON-SOLUTIONS 164 5.7
STABILITY FOR FAMILIES OF NON-LINEAR EQUATIONS 166
5.7.1 STABILITY FOR MONOTONE OPERATORS 167
5.7.2 STABILITY FOR WIDER CLASSES OF OPERATORS 171
5.7.3 TWO-STAGE SOLUTIONS 172
IMAGE 3
CONTENTS XIII
6 ORLICZ SPACES 175
6.1 YOUNG FUNCTIONS 175
6.2 MODULAR AND LUXEMBURG NORM 185
6.2.1 EXAMPLES OF ORLICZ SPACES 188
6.2.2 STRUCTURE OF ORLICZ SPACES 193
6.2.3 THE A 2 -CONDITION 197
6.3 PROPERTIES OF THE MODULAR 208
6.3.1 CONVERGENCE IN MODULAR 208
6.3.2 LEVEL SETS AND BALLS 211
6.3.3 BOUNDEDNESS OF THE MODULAR 212
7 ORLICZ NORM AND DUALITY 214
7.1 THE ORLICZ NORM 214
7.2 HOLDER'S INEQUALITY 215
7.3 LOWER SEMI-CONTINUITY AND DUALITY OF THE MODULAR 216
7.4 JENSEN'S INTEGRAL INEQUALITY AND THE CONVERGENCE IN MEASURE 220 7.5
EQUIVALENCE OF THE NORMS 224
7.6 DUALITY THEOREMS 227
7.7 REFLEXIVITY 233
7.8 SEPARABILITY AND BASES OF ORLICZ SPACES 234
7.8.1 SEPARABILITY 234
7.8.2 BASES 236
7.9 AMEMIYA FORMULA AND ORLICZ NORM 236
8 DIFFERENTIABILITY AND CONVEXITY IN ORLICZ SPACES 241
8.1 FLAT CONVEXITY AND WEAK DIFFERENTIABILITY 241
8.2 FLAT CONVEXITY AND GATEAUX DIFFERENTIABILITY OF ORLICZ SPACES . . .
244 8.3 ^4-DIFFERENTIABILITY AND 5-CONVEXITY 247
8.4 LOCAL UNIFORM CONVEXITY, STRONG SOLVABILITY AND FRECHET
DIFFERENTIABILITY OF THE CONJUGATE 253
8.4.1 E-SPACES 262
8.5 FRECHET DIFFERENTIABILITY AND LOCAL UNIFORM CONVEXITY IN ORLICZ
SPACES 267
8.5.1 FRECHET DIFFERENTIABILITY OF MODULAR AND LUXEMBURG NORM . . 267
8.5.2 FRECHET DIFFERENTIABILITY AND LOCAL UNIFORM CONVEXITY . . . 276
8.5.3 FRECHET DIFFERENTIABILITY OF THE ORLICZ NORM AND LOCAL UNIFORM
CONVEXITY OF THE LUXEMBURG NORM 278
8.5.4 SUMMARY 280
8.6 UNIFORM CONVEXITY AND UNIFORM DIFFERENTIABILITY 281
8.6.1 UNIFORM CONVEXITY OF THE ORLICZ NORM 283
8.6.2 UNIFORM CONVEXITY OF THE LUXEMBURG NORM 288
8.7 APPLICATIONS 293
IMAGE 4
XIV CONTENTS
8.7.1 REGULARIZATIONOFTIKHONOVTYPE 294
8.7.2 RITZ'S METHOD 298
8.7.3 A GREEDY ALGORITHM IN ORLICZ SPACE 299
9 VARIATIONAL CALCULUS 309
9.1 INTRODUCTION 309
9.1.1 EQUIVALENT VARIATIONAL PROBLEMS 311
9.1.2 PRINCIPLE OF POINTWISE MINIMIZATION 312
9.1.3 LINEAR SUPPLEMENT 313
9.2 SMOOTHNESS OF SOLUTIONS 316
9.3 WEAK LOCAL MINIMA 320
9.3.1 CARATHEODORY MINIMALE 325
9.4 STRONG CONVEXITY AND STRONG LOCAL MINIMA 325
9.4.1 STRONG LOCAL MINIMA 330
9.5 NECESSARY CONDITIONS 333
9.5.1 THE JACOBI EQUATION AS A NECESSARY CONDITION 333
9.6 C 1 -VARIATIONAL PROBLEMS 335
9.7 OPTIMAL PATHS 336
9.8 STABILITY CONSIDERATIONS FOR VARIATIONAL PROBLEMS 337
9.8.1 PARAMETRIC TREATMENT OF THE DIDO PROBLEM 339
9.8.2 DIDO PROBLEM 341
9.8.3 GLOBAL OPTIMAL PATHS 345
9.8.4 GENERAL STABILITY THEOREMS 346
9.8.5 DIDO PROBLEM WITH TWO-DIMENSIONAL QUADRATIC SUPPLEMENT . 349 9.8.6
STABILITY IN ORLICZ-SOBOLEV SPACES 352
9.9 PARAMETER-FREE APPROXIMATION OF TIME SERIES DATA BY MONOTONE
FUNCTIONS 354
9.9.1 PROJECTION ONTO THE POSITIVE CONE IN SOBOLEV SPACE 354 9.9.2
REGULARIZATIONOFTIKHONOV-TYPE 357
9.9.3 A ROBUST VARIANT 362
9.10 OPTIMAL CONTROL PROBLEMS 363
9.10.1 MINIMAL TIME PROBLEM AS A LINEAR L L -APPROXIMATION PROBLEM 365
BIBLIOGRAPHY 371
LIST OF SYMBOLS 379
INDEX 381 |
| any_adam_object | 1 |
| author | Kosmol, Peter 1942- Müller-Wichards, Dieter 1946- |
| author_GND | (DE-588)106833189 (DE-588)120759209 |
| author_facet | Kosmol, Peter 1942- Müller-Wichards, Dieter 1946- |
| author_role | aut aut |
| author_sort | Kosmol, Peter 1942- |
| author_variant | p k pk d m w dmw |
| building | Verbundindex |
| bvnumber | BV037266822 |
| classification_rvk | SK 600 SK 870 |
| ctrlnum | (OCoLC)711846808 (DE-599)DNB1007280395 |
| dewey-full | 519.76 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.76 |
| dewey-search | 519.76 |
| dewey-sort | 3519.76 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV037266822 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:00:49Z |
| institution | BVB |
| isbn | 9783110250206 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021179849 |
| oclc_num | 711846808 |
| open_access_boolean | |
| owner | DE-20 DE-11 DE-19 DE-BY-UBM DE-824 DE-83 DE-384 |
| owner_facet | DE-20 DE-11 DE-19 DE-BY-UBM DE-824 DE-83 DE-384 |
| physical | XIV, 388 S. Ill. - graph. Darst. 240 mm x 170 mm |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter series in nonlinear analysis and applications |
| series2 | De Gruyter series in nonlinear analysis and applications |
| spelling | Kosmol, Peter 1942- Verfasser (DE-588)106833189 aut Optimization in function spaces with stability considerations in Orlicz spaces Peter Kosmol ; Dieter Müller-Wichards Berlin [u.a.] De Gruyter 2011 XIV, 388 S. Ill. - graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter series in nonlinear analysis and applications 13 Banach-Raum (DE-588)4004402-6 gnd rswk-swf Konvexe Optimierung (DE-588)4137027-2 gnd rswk-swf Orlicz-Raum (DE-588)4172841-5 gnd rswk-swf Konvexe Optimierung (DE-588)4137027-2 s Banach-Raum (DE-588)4004402-6 s Orlicz-Raum (DE-588)4172841-5 s DE-604 Müller-Wichards, Dieter 1946- Verfasser (DE-588)120759209 aut Erscheint auch als Online-Ausgabe 978-3-11-025021-3 De Gruyter series in nonlinear analysis and applications 13 (DE-604)BV005530011 13 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3543530&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021179849&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kosmol, Peter 1942- Müller-Wichards, Dieter 1946- Optimization in function spaces with stability considerations in Orlicz spaces De Gruyter series in nonlinear analysis and applications Banach-Raum (DE-588)4004402-6 gnd Konvexe Optimierung (DE-588)4137027-2 gnd Orlicz-Raum (DE-588)4172841-5 gnd |
| subject_GND | (DE-588)4004402-6 (DE-588)4137027-2 (DE-588)4172841-5 |
| title | Optimization in function spaces with stability considerations in Orlicz spaces |
| title_auth | Optimization in function spaces with stability considerations in Orlicz spaces |
| title_exact_search | Optimization in function spaces with stability considerations in Orlicz spaces |
| title_full | Optimization in function spaces with stability considerations in Orlicz spaces Peter Kosmol ; Dieter Müller-Wichards |
| title_fullStr | Optimization in function spaces with stability considerations in Orlicz spaces Peter Kosmol ; Dieter Müller-Wichards |
| title_full_unstemmed | Optimization in function spaces with stability considerations in Orlicz spaces Peter Kosmol ; Dieter Müller-Wichards |
| title_short | Optimization in function spaces |
| title_sort | optimization in function spaces with stability considerations in orlicz spaces |
| title_sub | with stability considerations in Orlicz spaces |
| topic | Banach-Raum (DE-588)4004402-6 gnd Konvexe Optimierung (DE-588)4137027-2 gnd Orlicz-Raum (DE-588)4172841-5 gnd |
| topic_facet | Banach-Raum Konvexe Optimierung Orlicz-Raum |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3543530&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=021179849&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005530011 |
| work_keys_str_mv | AT kosmolpeter optimizationinfunctionspaceswithstabilityconsiderationsinorliczspaces AT mullerwichardsdieter optimizationinfunctionspaceswithstabilityconsiderationsinorliczspaces |