Abstract convexity and global optimization
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2000
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| Schriftenreihe: | Nonconvex optimization and its applications
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| 100 | 1 | |a Rubinov, Aleksandr M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Abstract convexity and global optimization |c by Alexander Rubinov |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 2000 | |
| 300 | |a XVIII, 490 S. |b graph. Darst. | ||
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| adam_text | IMAGE 1
ABSTRACT CONVEXITY
AND GLOBAL OPTIMIZATION
BY ALEXANDER RUBINOV SCHOOL OF INFORMATION TECHNOLOGY AND MATHEMATICAL
SCIENCES,
UNIVERSITY OF BALLARAT, VICTORIA, AUSTRALIA
II
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
IMAGE 2
CONTENTS
PREFACE XI
ACKNOWLEDGMENT XVII
1. AN INTRODUCTION TO ABSTRACT CONVEXITY 1
1.1 OVERVIEW 1
1.2 PRELIMINARIES 2
1.3 ABSTRACT CONVEX FUNCTIONS AND SETS 3
1.4 SUBDIFFERENTIABILITY 9
1.5 CONJUGATION 12
1.6 ABSTRACT CONCAVE FUNCTIONS AND INFIMA OF ABSTRACT CONVEX FUNCTIONS
13
2. ELEMENTS OF MONOTONIC ANALYSIS: IPH FUNCTIONS AND NORMAL SETS 15
2.1 INTRODUCTION 15
2.2 INCREASING POSITIVELY HOMOGENEOUS FUNCTIONS DEFINED ON POSITIVE
ORTHANT 18
2.2.1 OVERVIEW 18
2.2.2 PRELIMINARIES 18
2.2.3 IPH FUNCTIONS 19
2.2.4 MIN-TYPE FUNCTIONS AND IPH FUNCTIONS 23
2.2.5 ABSTRACT CONVEXITY WITH RESPECT TO THE SET OF MINTYPE FUNCTIONS 26
2.2.6 LEVEL SETS OF IPH FUNCTIONS 28
2.2.7 POLARITY FOR NORMAL SETS AND IPH FUNCTIONS 30
2.2.8 SUPPORT SETS 33
2.2.9 SUBDIFFERENTIAL 36
2.2.10 CONCAVITY OF THE POLAR FUNCTION 37
2.2.11 COMPARISON WITH CONVEX ANALYSIS 39
2.3 INCREASING POSITIVELY HOMOGENEOUS FUNCTIONS DEFINED ON THE
NON-NEGATIVE ORTHANT 42
2.3.1 OVERVIEW 42
2.3.2 PRELIMINARIES 43
V
IMAGE 3
VI ABSTRACT CONVEXITY
2.4
2.3.3 2.3.4 2.3.5 2.3.6
2.3.7 2.3.8
L-CONVEX FUNCTION SUPPORT SETS TWO KINDS OF NORMALITY PROPERTIES OF THE
SUPPORT SETS SUBDIFFERENTIALS OF IPH FUNCTIONS ABSTRACT CONCAVITY AND
SUPERDIFFERENTIALS BEST APPROXIMATION BY NORMAL SETS
2.4.1 2.4.2 2.4.3 2.4.4 2.4.5
OVERVIEW PRELIMINARIES DISTANCE TO A NORMAL SET
SEPARATION DISTANCE TO THE UNION AND THE INTERSECTION OF NORMAL SETS
44 46 47 50
54 57 60 60 61 63 66
71
3. ELEMENTS OF MONOTONIC ANALYSIS: MONOTONIC FUNCTIONS 75
3.1 INTRODUCTION 75
3.2 INCREASING CO-RADIANT FUNCTIONS 77
3.2.1 OVERVIEW 77
3.2.2 DEFINITION AND PROPERTIES OF ICR FUNCTIONS 77
3.2.3 ICR FUNCTIONS AND IPH FUNCTIONS 80
3.2.4 ABSTRACT CONVEXITY OF ICR FUNCTIONS 81
3.3 INCREASING CONVEX-ALONG-RAYS FUNCTIONS 82
3.3.1 OVERVIEW 82
3.3.2 ICAR FUNCTIONS: DEFINITION, EXAMPLES AND SOME PROPERTIES 83
3.3.3 ICAR FUNCTIONS AS ABSTRACT CONVEX FUNCTIONS 87
3.3.4 SUBDIFFERENTIABILITY OF ICAR FUNCTIONS 92
3.3.5 SUBDIFFERENTIABILITY OF STRICTLY ICAR FUNCTIONS 93
3.3.6 LIPSCHITZ FUNCTION AND ICAR FUNCTIONS 95
3.4 DECREASING FUNCTIONS 98
3.4.1 OVERVIEW 98
3.4.2 DECREASING FUNCTIONS AND IPH FUNCTIONS 98
3.4.3 MULTIPLICATIVE INF-CONVOLUTION 107
4. APPLICATION TO GLOBAL OPTIMIZATION: LAGRANGE AND PENALTY FUNCTIONS
113
4.1 INTRODUCTION 113
4.2 EXTENDED LAGRANGE AND PENALTY FUNCTIONS 117
4.2.1 OVERVIEW 117
4.2.2 PRELIMINARIES 117
4.2.3 EXTENDED LAGRANGE FUNCTIONS 119
4.2.4 EXTENDED PENALTY FUNCTIONS 123
4.2.5 EXAMPLES 124
4.2.6 SUPPORT SET OF THE DUAL FUNCTION 125
4.2.7 ANOTHER APPROACH 127
4.3 EXTENDED PENALIZATION FOR PROBLEMS WITH ONE CONSTRAINT 128 4.3.1
OVERVIEW 128
4.3.2 PRELIMINARIES 129
IMAGE 4
CONTENTS
VN
4.3.3 PERTURBATION FUNCTIONS 131
4.3.4 EXACT PENALIZATION 138
4.3.5 PENALIZATION BY IPH FUNCTIONS PK 144
5. ELEMENTS OF STAR-SHAPED ANALYSIS 153
5.1 INTRODUCTION 153
5.2 RADIANT AND CO-RADIANT SETS AND THEIR GAUGES 155
5.2.1 OVERVIEW 155
5.2.2 RADIANT AND CO-RADIANT SETS 155
5.2.3 RADIATIVE AND CO-RADIATIVE SETS 161
5.2.4 RADIANT SETS WITH LIPSCHITZ CONTINUOUS MINKOWSKI GAUGES 165
5.3 STAR-SHAPED SETS AND CO-STAR-SHAPED SETS 170
5.3.1 OVERVIEW 170
5.3.2 STAR-SHAPED SETS AND THEIR KERNEIS 170
5.3.3 SUM OF STAR-SHAPED SETS AND SUM OF CO-STAR-SHAPED SETS 175
5.4 SEPARATION 180
5.4.1 OVERVIEW 180
5.4.2 CONE-SEPARATION AND SEPARATION BY A FINITE COLLECTION OF LINEAR
FUNCTIONS 181
5.4.3 SEPARATION OF STAR-SHAPED SETS 186
5.4.4 SEPARATION OF CO-STAR-SHAPED SETS 191
5.5 ABSTRACT CONVEXITY WITH RESPECT TO GENERAL MIN-TYPE FUNCTIONS 202
5.5.1 OVERVIEW 202
5.5.2 POSITIVELY HOMOGENEOUS ABSTRACT CONVEX FUNCTIONS 203 5.5.3
%-CONVEX FUNCTIONS 211
5.5.4 SUBDIFFERENTIALS OF % N+ I-CONVEX FUNCTIONS 220
5.5.5 ABSTRACT CONVEX SETS 223
5.5.6 OTHER CLASSES OF ABSTRACT CONVEX FUNCTIONS 226
6. SUPREMAL GENERATORS AND THEIR APPLICATIONS 229 6.1 INTRODUCTION 229
6.2 CONTINUOUS AND LOWER SEMICONTINUOUS FUNCTIONS 231
6.2.1 OVERVIEW 231
6.2.2 LOWER SEMICONTINUOUS FUNCTIONS 231
6.2.3 EXAMPLES 237
6.2.4 ICAR EXTENSIONS OF FUNCTIONS DEFINED ON THE UNIT SIMPLEX 239
6.3 SUPREMAL GENERATORS FOR SPACES OF HOMOGENEOUS FUNCTIONS 240
6.3.1 OVERVIEW 240
6.3.2 PRELIMINARIES 241
6.3.3 HOMOGENEOUS FUNCTIONS OF DEGREE ONE 242
6.3.4 SYMMETRIE POSITIVELY HOMOGENEOUS FUNCTIONS OF DEGREE TWO 244
6.4 SOME APPLICATIONS OF SUPREMAL GENERATORS 247
IMAGE 5
VIII ABSTRACT CONVEXITY
6.4.1 OVERVIEW 247
6.4.2 CONVERGENCE OF SEQUENCES OF POSITIVE FUNCTIONALS 248 6.4.3 THE
SUPREMAL RANK OF A COMPACT SET 251
6.5 APPLICATION TO HADAMARD-TYPE INEQUALITIES 254
6.5.1 OVERVIEW 254
6.5.2 HADAMARD-TYPE INEQUALITIES FOR CONVEX FUNCTIONS 255 6.5.3
QUASICONVEX FUNCTIONS AND P-FUNCTIONS 256
6.5.4 INEQUALITIES OF HADAMARD TYPE FOR P-FUNCTIONS AND QUASICONVEX
FUNCTIONS 262
6.5.5 INEQUALITY OF HADAMARD TYPE FOR ICAR FUNCTIONS 265 7. FURTHER
ABSTRACT CONVEXITY 271
7.1 INTRODUCTION 271
7.2 ABSTRACT CONVEXITY WITH RESPECT ON A SUBSET 272
7.2.1 OVERVIEW 272
7.2.2 BASIC DEFINITIONS AND PROPERTIES 273
7.2.3 SUBDIFFERENTIABILITY 278
7.2.4 CONJUGATION AND APPROXIMATE SUBDIFFERENTIALS 282
7.2.5 ABSTRACT CONVEXITY AND GLOBAL MINIMIZATION 288
7.2.6 MINIMAX RESULT FOR ABSTRACT CONVEX FUNCTIONS 290
7.2.7 POSITIVELY HOMOGENEOUS FUNCTIONS 291
7.2.8 POSITIVELY HOMOGENEOUS EXTENSION 294
7.2.9 POLARITY FOR FUNCTIONS AND SETS, WHICH ARE ABSTRACT CONVEX WITH
RESPECT TO A CONIC SET OF POSITIVELY HOMOGENEOUS FUNCTIONS 297
7.3 SOME CLASSES OF ABSTRACT CONVEX FUNCTIONS 304
7.3.1 OVERVIEW 304
7.3.2 LINEAR FUNCTIONS - GENERATING SUBLINEARITY 304
7.3.3 AFFINE FUNCTIONS - GENERATING CONVEXITY 308
7.3.4 MIN-TYPE FUNCTIONS - GENERATING CONVEXITY-ALONGRAYS 315
7.3.5 TWO-STEP FUNCTIONS - GENERATING QUASICONVEXITY. ABSTRACT CONVEX
FUNCTIONS 316
7.3.6 TWO-STEP FUNCTIONS - GENERATING QUASICONVEXITY. ABSTRACT CONVEX
SETS 321
7.3.7 TWO-STEP FUNCTIONS - GENERATING QUASICONVEXITY. ABSTRACT
SUBDIFFERENTIALS 329
7.3.8 INFIMA OF FAMILIES OF ABSTRACT CONVEX FUNCTIONS 333
7.4 MINKOWSKI DUALITY, C 2 -LATTICES AND SEMILINEAR LATTICES 335
7.4.1 OVERVIEW 335
7.4.2 THE MINKOWSKI DUALITY 336
7.4.3 MINKOWSKI DUALITY FOR CMATTICES AND SEMILINEAR LATTICES 338
8. APPLICATION TO GLOBAL OPTIMIZATION: DUALITY 345 8.1 INTRODUCTION 345
8.2 GENERAL SOLVABILITY THEOREMS 347
8.2.1 OVERVIEW 347
IMAGE 6
CONTENTS
IX
8.2.2 SOLVABILITY THEOREMS FOR SYSTEMS OF ABSTRACT CONVEX FUNCTIONS 348
8.2.3 FURTHER SOLVABILITY RESULTS 350
8.2.4 SUBLINEAR INEQUALITY SYSTEMS 354
8.2.5 CONVEX INEQUALITY SYSTEMS 355
8.2.6 POSITIVELY HOMOGENEOUS SYSTEMS 356
8.2.7 TWICE CONTINUOUSLY DIFFERENTIABLE SYSTEMS 359
8.2.8 QUASICONVEX INEQUALITY SYSTEMS 360
8.2.9 SYSTEMS INVOLVING FUNCTIONS EXPRESSIBLE AS THE INFIMA OF FAMILIES
OF CONVEX FUNCTIONS 361
8.2.10 CONVEX MAXIMIZATION 362
8.3 MAXIMAL ELEMENTS OF SUPPORT SETS AND TOLAND-SINGER FORMULA 363
8.3.1 OVERVIEW 363
8.3.2 MAXIMAL ELEMENTS 364
8.3.3 EXISTENCE OF MAXIMAL ELEMENTS OF SUPPORT SETS 367
8.3.4 POSITIVELY HOMOGENEOUS ELEMENTARY FUNCTIONS 371
8.3.5 MAXIMAL ELEMENTS OF SUPPORT SETS WITH RESPECT TO CONIC SETS OF
POSITIVELY HOMOGENEOUS FUNCTIONS 373 8.3.6 THE TOLAND-SINGER FORMULA AND
MAXIMAL ELEMENTS OF SUPPORT SETS 375
8.3.7 EXCESS FUNCTIONS 378
8.4 OPTIMIZATION OF THE DIFFERENCE OF ABSTRACT CONVEX FUNCTIONS 380
8.4.1 OVERVIEW 380
8.4.2 MINIMIZATION OF THE DIFFERENCE OF COERCIVE CONVEX FUNCTIONS 382
8.4.3 MINIMIZATION OF THE DIFFERENCE OF COERCIVE CONVEX AND SUBLINEAR
FUNCTIONS 386
8.4.4 DUAL OPTIMALITY CONDITIONS FOR THE DIFFERENCE OF ICAR FUNCTIONS
391
8.4.5 NECESSARY AND SUFFICIENT CONDITIONS FOR THE MINIMUM OF THE
DIFFERENCE OF COERCIVE STRICTLY ICAR FUNCTIONS 393
8.4.6 MINIMIZATION OF THE QUOTIENT OF CONVEX FUNCTIONS 396
APPLICATION TO GLOBAL OPTIMIZATION: NUMERICAL METHODS 399
9.1 INTRODUCTION 399
9.2 CONCEPTUAL SCHEMES OF NUMERICAL METHODS 401
9.2.1 OVERVIEW 401
9.2.2 SPECIAL MINORATES OF ABSTRACT CONVEX FUNCTIONS 401
9.2.3 GENERALIZED CUTTING PLANE METHOD 404
9.2.4 BRANCH-AND-BOUND METHODS 410
9.2.5 TABU SEARCH 413
9.2.6 EXTERNAL CENTRES METHOD 415
9.2.7 LIPSCHITZ PROGRAMMING VIA ABSTRACT CONVEXITY 416
9.3 CUTTING ANGLE METHOD 418
9.3.1 OVERVIEW 418
IMAGE 7
X ABSTRACT CONVEXITY
9.3.2 9.3.3 9.3.4 9.3.5
9.3.6 9.3.7 9.3.8 9.3.9
CUTTING ANGLE METHOD FOR ICAR FUNCTIONS THE SUBPROBLEM NUMERICAL RESULTS
- ICAR OBJECTIVE FUNCTIONS CUTTING ANGLE METHOD FOR INCREASING
CO-RADIANT FUNCTIONS
NUMERICAL RESULTS - ICR OBJECTIVE FUNCTIONS CUTTING ANGLE METHOD FOR
LIPSCHITZ FUNCTIONS NUMERICAL RESULTS - LIPSCHITZ FUNCTIONS
BRANCH-AND-BOUND METHOD FOR LIPSCHITZ FUNCTIONS 9.4 CUTTING ANGLE METHOD
(CONTINUATION)
9.4.1 9.4.2
9.4.3 9.4.4 9.4.5 9.4.6
OVERVIEW CUTTING ANGLE METHOD FOR THE MINIMIZATION OF IPH FUNCTIONS OVER
THE UNIT SIMPLEX THE SUBPROBLEM: LOCAL MINIMA THE SUBPROBLEM: GLOBAL
MINIMA RESULTS OF NUMERICAL EXPERIMENTS AN EXACT METHOD FOR SOLVING THE
SUBPROBLEM 9.5 MONOTONE OPTIMIZATION
9.5.1 9.5.2 9.5.3 9.5.4 9.5.5 9.5.6 9.5.7
REFERENCES
OVERVIEW PRELIMINARIES PROBLEMS OF MONOTONIC OPTIMIZATION BASIC
PROPERTIES PROPOSED SOLUTION METHOD
CONVERGENCE COMPUTATIONAL EXPERIENCE
419 421 424
427 429 432 436 438 439 439
440 442 447 450 454
460 460 461 463 464 466 468 469 471
INDEX
489
|
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| id | DE-604.BV013409142 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T15:27:10Z |
| institution | BVB |
| isbn | 079236323X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009150149 |
| oclc_num | 43864387 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-706 DE-11 |
| owner_facet | DE-703 DE-824 DE-706 DE-11 |
| physical | XVIII, 490 S. graph. Darst. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Kluwer |
| record_format | marc |
| series | Nonconvex optimization and its applications |
| series2 | Nonconvex optimization and its applications |
| spellingShingle | Rubinov, Aleksandr M. Abstract convexity and global optimization Nonconvex optimization and its applications Convexe functies gtt Mathematische programmering gtt Optimaliseren gtt Convex programming Mathematical optimization Globale Optimierung (DE-588)4140067-7 gnd Konvexe Hülle (DE-588)4312637-6 gnd |
| subject_GND | (DE-588)4140067-7 (DE-588)4312637-6 |
| title | Abstract convexity and global optimization |
| title_auth | Abstract convexity and global optimization |
| title_exact_search | Abstract convexity and global optimization |
| title_full | Abstract convexity and global optimization by Alexander Rubinov |
| title_fullStr | Abstract convexity and global optimization by Alexander Rubinov |
| title_full_unstemmed | Abstract convexity and global optimization by Alexander Rubinov |
| title_short | Abstract convexity and global optimization |
| title_sort | abstract convexity and global optimization |
| topic | Convexe functies gtt Mathematische programmering gtt Optimaliseren gtt Convex programming Mathematical optimization Globale Optimierung (DE-588)4140067-7 gnd Konvexe Hülle (DE-588)4312637-6 gnd |
| topic_facet | Convexe functies Mathematische programmering Optimaliseren Convex programming Mathematical optimization Globale Optimierung Konvexe Hülle |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009150149&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010085908 |
| work_keys_str_mv | AT rubinovaleksandrm abstractconvexityandglobaloptimization |