Global optimization with non-convex constraints: sequential and parallel algorithms
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
2000
|
| Series: | Nonconvex optimization and its applications
45 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVIII, 699 S. graph. Darst. |
| ISBN: | 0792364902 |
Staff View
MARC
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Record in the Search Index
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| adam_text | IMAGE 1
GLOBAL OPTIMIZATION WITH
NON-CONVEX CONSTRAINTS SEQUENTIAL AND PARALLEL ALGORITHMS
BY
ROMAN G. STRONGIN
NIZHNI NOVGOROD STATE UNIVERSITY, NIZHNI NOVGOROD, RUSSIA
AND
YAROSLAV D. SERGEYEV
INSTITUTE OF SYSTEMS ANALYSIS AND INFORMATION TECHNOLOGY, UNIVERSITY OF
CALABRIA, RENDE, ITALY AND
NIZHNI NOVGOROD STATE UNIVERSITY, NIZHNI NOVGOROD, RUSSIA
W
KLUWER ACADEMIC PUBLISHERS DORDRECHT / BOSTON / LONDON
IMAGE 2
CONTENTS
P R E F A CE XVII
A C K N O W L E D G E M E N TS XXVII
PART ONE GLOBAL O P T I M I Z A T I ON A L G O R I T H MS AS D E C I S I
ON P R O C E D U R E S. T H E O R E T I C AL B A C K G R O U ND A ND C O
RE U N I V A R I A TE
CASE 1
1 I N T R O D U C T I ON 3
1.1 OPTIMIZATION PROBLEMS AND SEARCH TECHNIQUES 3
1.2 A PRIORI INFORMATION AND ESTIMATES FOR AN OPTIMUM 10
ROLE OF A PRIORI INFORMATION 10
UNIMODALITY AND LOCAL IMPROVEMENTS 11
MULTIMODALITY AND ADAPTATION OF LOCAL TECH NIQUES 13
A PRIORI INFORMATION AND EXPANSION IN STANDARD PROBLEMS 17
LIPSCHITZ CONTINUITY ASSUMPTIONS AND GLOBAL OPTIMALITY 19
OBJECTIVE FUNCTION AS A SAMPLE OF SOME RANDOM FUNCTION 25
1.3 DECISION RULES AS MINIMAX OPTIMAL STRATEGIES 28
MINIMAX APPROACH 29
ONE-STEP OPTIMALITY PRINCIPLE 33
VN
IMAGE 3
VIII G L O B AL O P T I M I Z A T I ON W I TH N O N - C O N V EX C O N S
T R A I N TS
1.4 INFORMATION-STATISTICAL APPROACH AND AVERAGE OPTIMALITY 35
1.5 PROBLEM OF DIMENSIONALITY AND REDUCTION TO ONE DI MENSION 40
EXPONENTIAL GROWTH OF THE GRID TECHNIQUE COMPLEXITY WITH THE INCREASE OF
DIMENSIONALITY 40 INCREASING COMPLEXITY OF BUILDING EFFECTIVE GRIDS IN
MANY DIMENSIONS 41
REDUCTION TO ONE DIMENSION 44
1.6 CONSTRAINTS AND REDUCTION TO UNCONSTRAINED CASE WITHOUT PENALTIES 47
OPTIMALITY AND CONSTRAINTS 47
PARTIAL COMPUTABILITY OF PROBLEM FUNCTIONALS 49
INDEXES AND COMPATIBILITY OF CONSTRAINTS 50
REDUCTION TO THE UNCONSTRAINED CASE 50
G L O B AL O P T I M I Z A T I ON A L G O R I T H MS AS S T A T I S T I
C AL D E C I S I ON P R O C E D U R ES - T HE I N F O R M A T I ON A P P
R O A CH 53
2.1 ESTIMATES FOR THE GLOBAL OPTIMUM BASED ON A STOCHASTIC DESCRIPTION
OF THE PROBLEM 53
A PRIORI DESCRIPTION AND ESTIMATES 53
MODEL FOR OUTCOMES OF TRIALS 55
A POSTERIORI ESTIMATES FOR THE GLOBAL OPTIMIZER 57
ESTIMATES FOR THE CASE OF ERROR-FREE OBSERVA T I ON 59
2.2 APPROXIMATE STOCHASTIC ESTIMATORS FOR THE GLOBAL OP TIMIZER 62
SIMPLIFIED ESTIMATORS FOR THE GLOBAL OPTIMIZER 62
SUFFICIENT CONDITIONS FOR APPROXIMATION 64
PARTICULAR STOCHASTIC MODEL 67
BAYESIAN ESTIMATES FOR ERROR-FREE OBSERVATIONS 70 BAYESIAN ESTIMATES FOR
OBSERVATIONS CORRUPTED BY GAUSSIAN ERRORS 74
IMAGE 4
CONTENTS
IX
2.3 DECISION AND TERMINATION RULES FOR ERROR-FREE OBSER VATIONS 78
FORECASTING OUTCOMES OF TRIALS 78
ONE-STEP OPTIMAL DECISIONS 82
TERMINATION RULE AND THE SEARCH ALGORITHM 84
RANDOMIZED STRATEGIES 85
2.4 DECISION RULES FOR OBSERVATIONS CORRUPTED BY ADDITIVE NON-BIASED
GAUSSIAN ERRORS 90
FORECASTING OUTCOMES OF NOISE-CORRUPTED TRIALS 90 DECISION RULE AND
CONVERGENCE STUDY 93
NUMERICAL SIMULATION OF SEARCH IN THE PRESENCE OF NOISE 98
2.5 ESTIMATIONS AND DECISIONS IN PROBLEMS OF EQUATION SOLVING 106
STOCHASTIC MODEL AND BAYESIAN ESTIMATES FOR A ROOT OF AN EQUATION 106
DECISION RULE AND CONVERGENCE STUDY 113
ROOT SEARCH ALGORITHMS 121
CORE GLOBAL SEARCH ALGORITHM AND CONVERGENCE STUDY 127
3.1 GLOBAL SEARCH ALGORITHM 127
3.2 CONVERGENCE CONDITIONS 133
LIPSCHITZIAN CASE 133
DISCONTINUOUS CASE 138
SMOOTHING GLOBAL SEARCH ALGORITHM 144
3.3 RATE OF CONVERGENCE 149
DENSITY OF TRIALS 149
SEQUENCE STRUCTURES IN THE RANGES OF FUNCTION LINEARITY 151
COMPARISON WITH THE GRID TECHNIQUE IN RANGES OF FUNCTION LINEARITY 160
3.4 TERMINATION CRITERION AND TRUNCATED SEQUENCES 162
3.5 MONOTONOUS CONVERGENCE 170
MONOTONOUS AND NEARLY MONOTONOUS CONVERGENCE 170
IMAGE 5
X G L O B AL O P T I M I Z A T I ON W I TH N O N - C O N V EX C O N S T
R A I N TS
CONDITIONS FOR NEARLY MONOTONOUS CONVERGENCE 176
MONOTONOUS GLOBAL SEARCH ALGORITHM 184
3.6 A PRIORI ASSUMPTIONS AND ACCELERATION OF SEARCH 193
LOCAL REFINEMENT OF THE BEST CURRENT ESTIMATE 193 OPERATING
CHARACTERISTICS AND THE DECISION RULES EFFICIENCY 202
FUNCTION RANGE GLOBAL SEARCH ALGORITHM 212
PROBLEMS WITH SIMPLE GEOMETRIE CONSTRAINTS 214
3.7 GLOBAL OPTIMIZATION OVER THE SET OF OPEN INTERVALS 216
ALGORITHM AND CONVERGENCE CONDITIONS 216
CASE OF MONOTONICALLY UNIMODAL CONSTRAINTS 223
G L O B AL O P T I M I Z A T I ON M E T H O DS AS B O U N D I NG P R O C
E D U R ES - T HE G E O M E T R IC A P P R O A CH 231
4.1 INTRODUCTION TO THE GEOMETRIE APPROACH 231
4.2 LOCAL TUNING OF BOUNDING PROCEDURES FOR LIPSCHITZ PROBLEMS 235
CONVERGENCE CONDITIONS 237
CONVERGENCE RATE AND STABILITY 241
NUMERICAL EXAMPLES 246
4.3 ALGORITHMS USING NON-SMOOTH BOUNDING PROCEDURES FOR PROBLEMS WITH
LIPSCHITZ DERIVATIVES 249
BOUNDING PROCEDURES WITH SIMPLE ESTIMATES OF THE LIPSCHITZ CONSTANT 249
USING LOCAL TUNING FOR OBTAINING MORE PRECISE NON-SMOOTH AUXILIARY
FUNCTIONS 252
CONVERGENCE CONDITIONS 258
4.4 METHODS USING SMOOTH BOUNDING PROCEDURES 266
SMOOTH BOUNDING PROCEDURES 266
A GENERAL DESCRIPTION OF THE METHODS USING SMOOTH BOUNDING PROCEDURES
270
ALGORITHM USING THE EXACT A PRIORI GIVEN GLOBAL LIPSCHITZ CONSTANT 274
IMAGE 6
CONTENTS XI
ALGORITHM ADAPTIVELY ESTIMATING THE GLOBAL LIPSCHITZ CONSTANT DURING THE
SEARCH 274
ALGORITHM ADAPTIVELY ESTIMATING THE LOCAL LIPSCHITZ CONSTANTS 277
CONVERGENCE CONDITIONS 278
NUMERICAL EXAMPLES 284
4.5 LOCAL TUNING AND THE RELATIONSHIP BETWEEN THE IN FORMATION AND
GEOMETRIE APPROACHES 288
CONVERGENCE CONDITIONS AND NUMERICAL EXAMPLES 291 4.6 FAST FINDING THE
FIRST ROOT OF AN EQUATION BY THE METHODS USING SMOOTH BOUNDING
PROCEDURES 295
FILTERS AS AN EXAMPLE OF APPLICATIONS WHERE THE PROBLEM ARISES 298
DESCRIPTION OF THE ALGORITHMS 302
CONVERGENCE ANALYSIS 309
NUMERICAL EXPERIMENTS 312
PART TWO G E N E R A L I Z A T I O NS F OR P A R A L L EL C O M P U T I
N G, C O N S T R A I N ED A ND M U L T I P LE C R I T E R IA P R O B L E
MS 317
5 P A R A L L EL G L O B AL O P T I M I Z A T I ON A L G O R I T H MS A
ND E V A L U A T I ON OF T HE E F F I C I E N CY OF P A R A L L E L I SM
319
5.1 FROM FAST SEQUENTIAL METHODS TOWARDS NON-REDUNDANT PARALLEL
ALGORITHMS 319
5.2 INFORMATION ALGORITHM WITH PARALLEL TRIALS 324
DECISION RULES OF THE INFORMATION ALGORITHM WITH PARALLEL TRIALS 325
CONVERGENCE CONDITIONS 328
ESTIMATES OF THE EFFICIENCY OF PARALLELISM 332
5.3 PARALLEL METHOD FOR SOLVING PROBLEMS WITH THE OBJEETIVE FUNCTIONS
SATISFYING A GENERALIZED LIPSCHITZ CONDITION 343
DECISION RULES OF THE METHOD 344
IMAGE 7
XII G L O B AL O P T I M I Z A T I ON W I TH N O N - C O N V EX C O N S
T R A I N TS
CONVERGENCE CONDITIONS OF THE PARALLEL ALGORITHM 347
EFFICIENCY EVALUATION 352
NUMERICAL EXAMPLES 359
5.4 PARALLEL ALGORITHM FOR SOLVING PROBLEMS WITH LIPSCHITZ DERIVATIVES
360
DESCRIPTION OF THE ALGORITHM 361
CONVERGENCE CONDITIONS 363
EFFICIENCY OF PARALLELIZATION 368
NUMERICAL EXAMPLES 375
G L O B AL O P T I M I Z A T I ON U N D ER N O N - C O N V EX C O N S T
R A I N TS - T HE I N D EX A P P R O A CH 379
6.1 PROBLEMS WITH PARTIALLY DEFINED OBJECTIVE FUNCTION AND CONSTRAINTS
379
6.2 REDUCTION TO CORE UNCONSTRAINED PROBLEM 382
6.3 INDEX METHOD OF GLOBAL OPTIMIZATION 385
6.4 CONVERGENCE CONDITIONS 391
6.5 E-RESERVED SOLUTIONS AND ACCELERATION OF SEARCH 396
-RESERVED SOLUTIONS AND CONVERGENCE PROPERTIES 396
RESERVES AND THE RATE OF CONVERGENCE 403
INDEX METHOD WITH ADAPTIVE RESERVES 405
6.6 LOCAL TUNING FOR SOLVING PROBLEMS WITH NON-CONVEX CONSTRAINTS 409
DESCRIPTION OF THE ALGORITHM 410
SUFFICIENT CONDITIONS OF GLOBAL CONVERGENCE 413
NUMERICAL EXPERIMENTS 415
A L G O R I T H MS F OR M U L T I P LE C R I T E R IA M U L T I E X T R
E M AL P R O B L E MS 419
7.1 MULTIOBJECTIVE OPTIMIZATION AND SCALARIZATION TECHNIQUES 419
STATEMENT OF THE PROBLEM 419
SCALARIZATION TECHNIQUE 421
IMAGE 8
CONTENTS XLLL
7.2 GLOBAL SEARCH ALGORITHM FOR MULTICRITERIA PROBLEMS 424
7.3 MULTIPLE CRITERIA PROBLEMS WITH NON-CONVEX CONSTRAINTS 431
PART T H R EE G L O B AL O P T I M I Z A T I ON IN M A NY D I M E N S I
O N S. G E N E R A L I Z A T I O NS T H R O U GH P E A NO C U R V ES 443
8 P E A N O - T Y PE S P A C E - F I L L I NG C U R V ES AS M E A NS F
OR M U L T I V A R I A TE P R O B L E MS 445
8.1 PEANO CURVES AND MULTIDIMENSIONAL GLOBAL OPTIMIZATION 445
SPACE-FILLING CURVES AND REDUCTION OF DIMENSIONALITY 445
ALGORITHM FOR UNCONSTRAINED GLOBAL SEARCH IN MANY DIMENSIONS 453
LOCAL REFINEMENT OF THE BEST CURRENT ESTIMATES 461 OPTIMIZATION OVER THE
CUBE WITH CAVITIES BUILT OF SUBCUBES 463
SEARCH FOR THE GLOBAL MINIMIZER YIELDING THE KNOWN OPTIMAL VALUE 464
8.2 APPROXIMATIONS TO PEANO CURVES 467
ADJACENT SUBCUBES 467
NUMERATION IN THE FIRST PARTITION 468
NUMERATION IN THE SECOND PARTITION 471
LINKING NUMERATIONS IN SUBSEQUENT PARTITIONS 475 APPROXIMATION BY
CENTERS OF THE M TH PARTITION SUBCUBES 484
PIECEWISE-LINEAR APPROXIMATIONS TO PEANO CURVES 485
PEANO CURVES VERSUS SPIRALS AND TV EVOLVENTS 490 NON-UNIVALENT
PEANO-LIKE EVOLVENTS 492
STANDARD ROUTINES FOR COMPUTING APPROXIMA TIONS TO PEANO CURVES 499
8.3 INDEX SCHEME FOR MULTIDIMENSIONAL CONSTRAINED PROB LEMS 511
IMAGE 9
G L O B AL O P T I M I Z A T I ON W I TH N O N - C O N V EX C O N S T R
A I N TS
REDUCTION TO ONE DIMENSION 511
MULTIVARIATE INDEX METHOD 513
CONVERGENCE CONDITIONS 523
8.4 MULTICRITERIA SCHEME IN MANY DIMENSIONS 531
8.5 PEANO CURVES AND LOCAL TUNING FOR SOLVING MULTIDI MENSIONAL
PROBLEMS 541
M U L T I D I M E N S I O N AL P A R A L L EL A L G O R I T H MS 551
9.1 PARALLEL MULTIDIMENSIONAL INFORMATION ALGORITHM 552
9.2 PARALLEL MULTIDIMENSIONAL INFORMATION ALGORITHM WITH ADAPTIVE LOCAL
TUNING 558
9.3 PARALLEL CHARACTERISTIC ALGORITHMS 566
CLASS OF PARALLEL CHARACTERISTIC GLOBAL OPTIMIZ ATION ALGORITHMS 566
CONVERGENCE OF PARALLEL CHARACTERISTIC ALGORITHMS 569
CONDITIONS OF NON-REDUNDANT PARALLELIZATION 580
NUMERICAL EXAMPLES 586
9.4 PARALLEL ASYNCHRONOUS GLOBAL SEARCH AND THE NESTED OPTIMIZATION
SCHEME 590
NESTED OPTIMIZATION SCHEME AND PARALLEL COMPUTATIONS 590
ASYNCHRONOUS PARALLEL ALGORITHM FOR UNIVARIATE GLOBAL OPTIMIZATION
PROBLEMS 595
CONVERGENCE AND NON-REDUNDANCY CONDITIONS 597 NUMERICAL EXAMPLES 604
M U L T I P LE P E A NO S C A N N I N GS A ND M U L T I D I M E N S I O
N AL P R O B L E MS 611
10.1 METRIE PROPERTIES IN ONE AND MANY DIMENSIONS: MUL TIPLE SHIFTED
SCANNINGS 611
REDUCTION TO ONE DIMENSION AND RETAINING THE PROPERTY OF NEARNESS 611
MULTIPLE SCANNING 613
IMAGE 10
CONTENTS
XV
METRIE PROPERTIES OF MULTIPLE SCANNINGS 616
10.2 ALGORITHM FOR GLOBAL MULTIDIMENSIONAL CONSTRAINED PROBLEMS
EMPLOYING MULTIPLE SCANNINGS 621
INDEX METHOD WITH MULTIPLE SCANNINGS 621
CONVERGENCE PROPERTIES 627
10.3 IMPLEMENTATION OF GLOBAL OPTIMIZATION SCHEMES WITH MULTIPLE
SCANNINGS ON MULTIPROCESSOR SYSTEMS 633
REDUCTION TO A FAMILY OF LINKED UNIVARIATE PROB LEMS 633
PARALLEL SCHEME AND SEARCH ALGORITHM 635
CONVERGENCE CONDITIONS 642
R E F E R E N C ES 651
LIST OF A L G O R I T H MS 679
LIST OF F I G U R ES 683
LIST OF T A B L ES 693
INDEX 697
|
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| id | DE-604.BV013891161 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:53:54Z |
| institution | BVB |
| isbn | 0792364902 |
| language | English |
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| series2 | Nonconvex optimization and its applications |
| spelling | Strongin, Roman G. Verfasser aut Global optimization with non-convex constraints sequential and parallel algorithms by Roman G. Strongin and Yaroslav D. Sergeyev Dordrecht [u.a.] Kluwer 2000 XXVIII, 699 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Nonconvex optimization and its applications 45 Mathematical optimization Nonconvex programming Globale Optimierung (DE-588)4140067-7 gnd rswk-swf Nichtkonvexe Optimierung (DE-588)4309215-9 gnd rswk-swf Globale Optimierung (DE-588)4140067-7 s Nichtkonvexe Optimierung (DE-588)4309215-9 s DE-604 Sergeev, Jaroslav D. Verfasser aut Nonconvex optimization and its applications 45 (DE-604)BV010085908 45 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009504405&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Strongin, Roman G. Sergeev, Jaroslav D. Global optimization with non-convex constraints sequential and parallel algorithms Nonconvex optimization and its applications Mathematical optimization Nonconvex programming Globale Optimierung (DE-588)4140067-7 gnd Nichtkonvexe Optimierung (DE-588)4309215-9 gnd |
| subject_GND | (DE-588)4140067-7 (DE-588)4309215-9 |
| title | Global optimization with non-convex constraints sequential and parallel algorithms |
| title_auth | Global optimization with non-convex constraints sequential and parallel algorithms |
| title_exact_search | Global optimization with non-convex constraints sequential and parallel algorithms |
| title_full | Global optimization with non-convex constraints sequential and parallel algorithms by Roman G. Strongin and Yaroslav D. Sergeyev |
| title_fullStr | Global optimization with non-convex constraints sequential and parallel algorithms by Roman G. Strongin and Yaroslav D. Sergeyev |
| title_full_unstemmed | Global optimization with non-convex constraints sequential and parallel algorithms by Roman G. Strongin and Yaroslav D. Sergeyev |
| title_short | Global optimization with non-convex constraints |
| title_sort | global optimization with non convex constraints sequential and parallel algorithms |
| title_sub | sequential and parallel algorithms |
| topic | Mathematical optimization Nonconvex programming Globale Optimierung (DE-588)4140067-7 gnd Nichtkonvexe Optimierung (DE-588)4309215-9 gnd |
| topic_facet | Mathematical optimization Nonconvex programming Globale Optimierung Nichtkonvexe Optimierung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009504405&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010085908 |
| work_keys_str_mv | AT stronginromang globaloptimizationwithnonconvexconstraintssequentialandparallelalgorithms AT sergeevjaroslavd globaloptimizationwithnonconvexconstraintssequentialandparallelalgorithms |