Separable programming: theory and methods
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
2001
|
| Series: | Applied optimization
53 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 314 S. |
| ISBN: | 0792368827 |
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MARC
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| adam_text | IMAGE 1
SEPARABLE PROGRAMMING
THEORY AND METHODS
BY STEFAN M. STEFANOV
DEPARTMENT OFMATHEMATICS, SOUTH-WEST UNIVERSITY, BLAGOEVGRAD, BULGARIA
M
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
IMAGE 2
CONTENTS
LIST OF FIGURES XI
LIST OF TABLES XIII
PREFACE XV
1 PRELIMINARIES: CONVEX ANALYSIS AND CONVEX PRO GRAMMING 1
1.1 CONVEX SETS. SUM OF SETS AND PRODUCT OF A SET
WITH A REAL NUMBER 1
1.2 PROJECTION OF A POINT ONTO A SET. SEPARATION OF
SETS AND SUPPORT OF SETS 8
1.3 CONVEX FUNCTIONS 15
1.4 DIRECTIONAL DERIVATIVES, SUBGRADIENTS AND SUBDIFFERENTIALS.
DIFFERENTIABLE CONVEX FUNCTIONS . . .. 24
1.5 CONVEX PROGRAMMING 41
1.6 LAGRANGIAN DUALITY 53
NOTES TO CHAPTER ONE 62
PART O NE - S E P A R A B LE P R O G R A M M I NG . 63 2 INTRODUCTION.
APPROXIMATING THE SEPARABLE PROB LEM 65
3 CONVEX SEPARABLE PROGRAMMING 79
4 SEPARABLE PROGRAMMING: A DYNAMIC PROGRAM MING APPROACH 91
4.1 THE DISCRETE GASE 91
4.2 FORWARD AND BACKWARD RECURSION. SUCCESSIVE DECISION MAKING 98
4.3 THE CONTINUOUS GASE 106
VII
IMAGE 3
VIII
SEPARABLE PROGRAMMING
4.4 MODELS INVOLVING TWO TYPES OF CONSTRAINTS. PROB LEM OF
DIMENSIONALITY IN DYNAMIC PROGRAMMING . 108 4.5 TWO-DIMENSIONAL
ALLOCATION PROBLEM 111
4.6 APPLICATION OF LAGRANGE MULTIPLIERS METHOD FOR REDUCING THE
DIMENSIONALITY OF THE PROBLEM . . .112
4.7 APPLICATION OF DYNAMIC PROGRAMMING APPROACH TO THE TRANSPORTATION
PROBLEM WITH NONLINEAR SEPARABLE OBJECTIVE FUNCTION 116
4.8 REVIEW OF SOME SEPARABLE INVENTORY AND OTHER MODELS 120
4.8.1 DETERMINISTIC STATIC MODELS 122
4.8.2 DYNAMIC INVENTORY MODELS : 128
4.8.3 PROBABILISTIC INVENTORY MODELS. NO-SETUP MODELS 134
4.8.4 INVESTMENT MODELS 137
NOTES TO PART ONE 140
PART T WO - C O N V EX S E P A R A B LE P R O G R A M M I NG W I TH B O
U N DS ON THE VARIABLES . 141 5 STATEMENT OF THE MAIN PROBLEM. BASIC
RESULT . . . 143
6 VERSION ONE: LINEAR EQUALITY CONSTRAINTS 151
6.1 SINGLE LINEAR EQUALITY CONSTRAINT 151
6.2 SEVERAL LINEAR EQUALITY CONSTRAINTS 154
7 THE ALGORITHMS 159
7.1 ANALYSIS OF THE SOLUTION TO PROBLEM (C) 159
7.2 STATEMENT OF ALGORITHM 1 (FOR (C)) 162
7.3 CONVERGENCE AND COMPUTATIONAL COMPLEXITY OF ALGORITHM 1 163
7.4 ANALYSIS OF THE SOLUTION TO PROBLEM ( C = ) 168
7.5 ALGORITHM 2 (FOR (C~)) AND ITS CONVERGENCE . . .. 170
7.6 COMMENTARY 173
8 VERSION TWO: LINEAR CONSTRAINT OF THE FORM 175
8.1 STATEMENT OF PROBLEM ( C -) AND RESULTS 175
8.2 ALGORITHM 3 (FOR PROBLEM (C^)) 179
IMAGE 4
CONTENTS
IX
9 WELL-POSEDNESS OF OPTIMIZATION PROBLEMS. ON
THE STABILITY OF THE SET OF SADDLE POINTS OF THE
LAGRANGIAN 181
9.1 WELL-POSEDNESS OF OPTIMIZATION PROBLEMS 181
9.1.1 TYCHONOV AND HADAMARD WELL-POSEDNESS. WELL-POSEDNESS IN THE
GENERALIZED SENSE 182 9.1.2 CALMNESS IN THE SENSE OF CLARKE 184
9.1.3 WELL-POSEDNESS OF PROBLEMS (C), (C=) AND (C^) 187
9.2 ON THE STABILITY OF THE SET OF SADDLE POINTS OF THE LAGRANGIAN 191
9.2.1 THE CONCEPT OF STABILITY OF SADDLE POINTS OF THE LAGRANGIAN 191
9.2.2 UNIMPORTANCE OF STABILITY OF THE SET OF SADDLE POINTS FOR THE
APPROACH SUGGESTED IN THIS WORK 193
10 EXTENSIONS 195
10.1 THEORETICAL ASPECTS 195
10.2 COMPUTATIONAL ASPECTS 199
11 APPLICATIONS AND COMPUTATIONAL EXPERIMENTS . . . 207 11.1 SOME
IMPORTANT FORMS OF FUNCTIONS FOR PROBLEMS (C),(C=) AND (C^) 207
11.2 COMPUTATIONAL EXPERIMENTS 221
NOTES TO PART TWO 223
PART T H R EE - SELECTED S U P P L E M E N T A RY TOPICS A ND
APPLICATIONS 227
12 APPROXIMATIONS WITH RESPECT TO T - AND ^ - N O R M S: AN APPLICATION
OF CONVEX SEPARABLE UNCONSTRAINED NONDIFFERENTIABLE OPTIMIZATION 229
12.1 INTRODUCTION. STATEMENT OF PROBLEMS UNDER CONSIDERATION 230
12.2 SOME PROPERTIES OF OBJECTIVE FUNCTIONS AND SOLVABILITY OF PROBLEMS
233
12.3 METHODS FOR SOLVING PROBLEMS UNDER CONSIDERATION236
IMAGE 5
X SEPARABLE PROGRAMMING
12.3.1 THEORETICAL MATTERS. THE SUBGRADIENT METHOD 236
12.3.2 CALCULATION OF SUBGRADIENTS 237
12.3.3 THE GRADIENT METHOD FOR DIFFERENTIABLE FUNCTIONS . . . 239
12.4 COMPUTATIONAL EXPERIMENTS AND CONCLUSIONS . . . 247 13 ABOUT
PROJECTIONS IN THE IMPLEMENTATION OF
STOCHASTIC QUASIGRADIENT METHODS TO SOME PROBABILISTIC INVENTORY CONTROL
PROBLEMS. THE STOCHAS TIC PROBLEM OF BEST CHEBYSHEV APPROXIMATION . .
251 13.1 INTRODUCTION 252
13.2 STOCHASTIC QUASIGRADIENT METHODS 254
13.3 ON THE PROJECTION IN THE IMPLEMENTATION OF
STOCHASTIC QUASIGRADIENT METHODS TO SOME PROBABILISTIC INVENTORY CONTROL
PROBLEMS. COMPUTA TIONAL EXPERIMENTS 259
13.4 THE STOCHASTIC PROBLEM OF BEST CHEBYSHEV AP PROXIMATION 262
14 INTEGRALITY OF THE KNAPSACK POLYTOPE 263
NOTES TO PART THREE 267
A P P E N D I C ES 269
A APPENDIX A - SOME DEFINITIONS AND THEOREMS
FROM CALCULUS 271
B APPENDIX B - METRIE, BANACH AND HUBERT SPACES 275 C APPENDIX C -
EXISTENCE OF SOLUTIONS TO OPTIMIZATION PROBLEMS - A GENERAL APPROACH 279
D APPENDIX D - BEST APPROXIMATION: EXISTENCE AND UNIQUENESS 285
B I B L I O G R A P H Y, I N D E X, NOTATION, LIST OF STATEMENTS 289
BIBLIOGRAPHY 291
INDEX 301
NOTATION 305
LIST OF STATEMENTS 309
|
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| dewey-raw | 519.7/6 |
| dewey-search | 519.7/6 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:55:12Z |
| institution | BVB |
| isbn | 0792368827 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009556417 |
| oclc_num | 46456402 |
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| owner_facet | DE-703 DE-11 |
| physical | XIX, 314 S. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | Kluwer |
| record_format | marc |
| series | Applied optimization |
| series2 | Applied optimization |
| spelling | Stefanov, Stefan M. Verfasser aut Separable programming theory and methods by Stefan M. Stefanov Dordrecht [u.a.] Kluwer 2001 XIX, 314 S. txt rdacontent n rdamedia nc rdacarrier Applied optimization 53 Convex programming Separable Optimierung (DE-588)4735512-8 gnd rswk-swf Separable Optimierung (DE-588)4735512-8 s DE-604 Applied optimization 53 (DE-604)BV010841718 53 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009556417&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stefanov, Stefan M. Separable programming theory and methods Applied optimization Convex programming Separable Optimierung (DE-588)4735512-8 gnd |
| subject_GND | (DE-588)4735512-8 |
| title | Separable programming theory and methods |
| title_auth | Separable programming theory and methods |
| title_exact_search | Separable programming theory and methods |
| title_full | Separable programming theory and methods by Stefan M. Stefanov |
| title_fullStr | Separable programming theory and methods by Stefan M. Stefanov |
| title_full_unstemmed | Separable programming theory and methods by Stefan M. Stefanov |
| title_short | Separable programming |
| title_sort | separable programming theory and methods |
| title_sub | theory and methods |
| topic | Convex programming Separable Optimierung (DE-588)4735512-8 gnd |
| topic_facet | Convex programming Separable Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009556417&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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