Stochastic optimization methods:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2005
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 301 - 308 Auch als Internetausgabe |
| Physical Description: | XIII, 314 S. graph. Darst. |
| ISBN: | 3540222723 9783540222729 |
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| 245 | 1 | 0 | |a Stochastic optimization methods |c Kurt Marti |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2005 | |
| 300 | |a XIII, 314 S. |b graph. Darst. | ||
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| 500 | |a Literaturverz. S. 301 - 308 | ||
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| 650 | 4 | |a Mathematical optimization | |
| 650 | 4 | |a Stochastic processes | |
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Record in the Search Index
| _version_ | 1807500524003524608 |
|---|---|
| adam_text |
CONTENTS
PAR
T I BASI
C STOCHASTI
C OPTIMIZATIO
N METHOD
S
1 DECISION/CONTRO
L UNDE
R STOCHASTI
C UNCERTAINT
Y
3
1.1 INTRODUCTION 3
1.2 DETERMINISTIC SUBSTITUTE PROBLEMS: BASIC FORMULATION 5
1.2.1 MINIMUM OR BOUNDED EXPECTED COSTS 6
1.2.2 MINIMUM OR BOUNDED MAXIMUM COSTS (WORST CASE) . 8
2 DETERMINISTI
C SUBSTITUT
E PROBLEM
S I
N OPTIMA
L DECISIO
N
UNDE
R STOCHASTI
C UNCERTAINT
Y
9
2.1 OPTIMUM DESIGN PROBLEMS WITH RANDOM PARAMETERS 9
2.1.1 DETERMINISTIC SUBSTITUTE PROBLEMS IN OPTIMAL DESIGN . 13
2.1.2 DETERMINISTIC SUBSTITUTE PROBLEMS IN QUALITY
ENGINEERING 16
2.2 BASIC PROPERTIES OF SUBSTITUTE PROBLEMS 18
2.3 APPROXIMATIONS OF DETERMINISTIC SUBSTITUTE PROBLEMS IN
OPTIMAL DESIGN 20
2.3.1 APPROXIMATION OF THE LOSS FUNCTION 20
2.3.2 REGRESSION TECHNIQUES, RSM 22
2.3.3 TAYLOR EXPANSION METHODS 25
2.4 APPLICATIONS TO PROBLEMS IN QUALITY ENGINEERING 28
2.5 APPROXIMATION OF PROBABILITIES - PROBABILITY INEQUALITIES 29
2.5.1 BONFERRONI-TYPE INEQUALITIES 30
2.5.2 TSCHEBYSCHEFF-TYPE INEQUALITIES 31
2.5.3 FORM (FIRST ORDER RELIABILITY METHODS) 36
2.6 CONSTRUCTION OF STATE FUNCTIONS IN STRUCTURAL ANALYSIS AND
DESIGN 38
2.6.1 PLASTIC ANALYSIS AND OPTIMAL PLASTIC DESIGN 38
2.6.2 OPTIMAL ELASTIC DESIGN 41
X CONTENTS
PAR
T I
I DIFFERENTIATIO
N METHOD
S
DIFFERENTIATIO
N METHOD
S FOR PROBABILIT
Y AND RISK FUNCTION
S
45
3.1 INTRODUCTION 45
3.2 TRANSFORMATION METHOD: DIFFERENTIATION BY USING AN INTEGRAL
TRANSFORMATION 48
3.2.1 REPRESENTATION OF THE DERIVATIVES BY SURFACE INTEGRALS.
. 53
3.3 THE DIFFERENTIATION OF STRUCTURAL RELIABILITIES 56
3.4 EXTENSIONS 58
3.4.1 MORE GENERAL RESPONSE (STATE) FUNCTIONS 58
3.5 COMPUTATION OF PROBABILITIES AND ITS DERIVATIVES BY
ASYMPTOTIC EXPANSIONS OF INTEGRAL OF LAPLACE TYPE 63
3.5.1 COMPUTATION OF STRUCTURAL RELIABILITIES AND ITS
SENSITIVITIES 63
3.5.2 NUMERICAL COMPUTATION OF DERIVATIVES OF TH
E
PROBABILITY FUNCTIONS ARISING IN CHANCE CONSTRAINED
PROGRAMMING 67
3.6 INTEGRAL REPRESENTATIONS OF THE PROBABILITY FUNCTION
P(X)
AND ITS DERIVATIVES 73
3.7 ORTHOGONAL FUNCTION SERIES EXPANSIONS I: EXPANSIONS IN
HERMITE FUNCTIONS, CASE M = 1 76
3.7.1 INTEGRALS OVER THE BASIS FUNCTIONS AND THE COEFFICIENTS
OF TH
E ORTHOGONAL SERIES 80
3.7.2 ESTIMATION/APPROXIMATION OF
P(X)
AND ITS DERIVATIVES . 83
3.7.3 THE INTEGRATED SQUARE ERROR (ISE) OF DETERMINISTIC
APPROXIMATIONS 89
3.8 ORTHOGONAL FUNCTION SERIES EXPANSIONS II: EXPANSIONS IN
HERMITE FUNCTIONS, CASE M 1 90
3.9 ORTHOGONAL FUNCTION SERIES EXPANSIONS
III:
EXPANSIONS IN
TRIGONOMETRIC, LEGENDRE AND LAGUERRE SERIES 92
3.9.1 EXPANSIONS IN TRIGONOMETRIC AND LEGENDRE SERIES 92
3.9.2 EXPANSIONS IN LAGUERRE SERIES 93
PART II
I DETERMINISTI
C DESCEN
T DIRECTION
S
4 DETERMINISTI
C DESCEN
T DIRECTION
S AND EFFICIENT POINT
S
97
4.1 CONVEX APPROXIMATION 97
4.1.1 APPROXIMATIVE CONVEX OPTIMIZATION PROBLEM 101
4.2 COMPUTATION OF DESCENT DIRECTIONS IN CASE OF NORMAL
DISTRIBUTIONS 103
4.2.1 DESCENT DIRECTIONS OF CONVEX PROGRAMS 107
4.2.2 SOLUTION OF THE AUXILIARY PROGRAMS 110
4.3 EFFICIENT SOLUTIONS (POINTS) 115
CONTENTS XI
4.3.1 NECESSAR
Y OPTIMALIT
Y CONDITION
S WITHOU
T GRADIENT
S .
. 117
4.3.2 EXISTENC
E OF FEASIBL
E DESCENT DIRECTION
S IN
NON-EFFICIENT SOLUTION
S OF (4.9A,B
) 119
4.4 DESCEN
T DIRECTION
S IN CASE OF ELLIPTICALL
Y CONTOURE
D
DISTRIBUTION
S 119
4.5 CONSTRUCTIO
N OF DESCENT DIRECTION
S BY USING QUADRATI
C
APPROXIMATION
S OF TH
E LOSS FUNCTIO
N 122
PAR
T I
V SEMI-STOCHASTI
C APPROXIMATIO
N METHOD
S
5 RSM-BASED STOCHASTI
C GRADIENT PROCEDURE
S
129
5.1 INTRODUCTION 129
5.2 GRADIENT ESTIMATION USING THE RESPONSE SURFACE
METHODOLOGY (RSM) 131
5.2.1 THE TWO PHASES OF RSM 134
5.2.2 THE MEAN SQUARE ERROR OF THE GRADIENT ESTIMATOR .
. 138
5.3 ESTIMATION OF THE MEAN SQUARE (MEAN FUNCTIONAL) ERROR 142
5.3.1 THE ARGUMENT CASE 143
5.3.2 THE CRITERIA! CASE 147
5.4 CONVERGENCE BEHAVIOR OF HYBRID STOCHASTIC APPROXIMATION
METHODS 147
5.4.1 ASYMPTOTICALLY CORRECT RESPONSE SURFACE MODEL 147
5.4.2 BIASED RESPONSE SURFACE MODEL 150
5-5 CONVERGENCE RATES OF HYBRID STOCHASTIC APPROXIMATION
PROCEDURES 153
5.5.1 FIXED RATE OF STOCHASTIC AND DETERMINISTIC STEPS 158
5.5.2 LOWER BOUNDS FOR THE MEAN SQUARE ERROR 168
5.5.3 DECREASING RATE OF STOCHASTIC STEPS 172
6 STOCHASTI
C APPROXIMATIO
N METHOD
S WIT
H CHANGIN
G ERROR
VARIANCES
177
6.1 INTRODUCTION 177
6.2 SOLUTION OF OPTIMALITY CONDITIONS 178
6.3 GENERAL ASSUMPTIONS AND NOTATIONS 179
6.3.1 INTERPRETATION OF THE ASSUMPTIONS 181
6.3.2 NOTATIONS AND ABBREVIATIONS IN THIS CHAPTER 182
6.4 PRELIMINARY RESULTS 183
6.4.1 ESTIMATION OF THE QUADRATIC ERROR 183
6.4.2 CONSIDERATION OF THE WEIGHTED ERROR SEQUENCE 185
6.4.3 FURTHER PRELIMINARY RESULTS 188
6.5 GENERAL CONVERGENCE RESULTS 190
6.5.1 CONVERGENCE WITH PROBABILITY ONE 190
6.5.2 CONVERGENCE IN THE MEAN 192
6.5.3 CONVERGENCE IN DISTRIBUTION 195
XII CONTENTS
6.6 REALISATION OF SEARCH DIRECTIONS
Y
N
204
6.6.1 ESTIMATION OF
G*
208
6.6.2 UPDATE OF THE JACOBIAN 209
6.6.3 ESTIMATION OF ERROR VARIANCES 214
6.7 REALIZATION OF ADAPTIVE STEP SIZES 219
6.7.1 OPTIMAL MATRIX STEP SIZES 220
6.7.2 ADAPTIVE SCALAR STEP SIZE 226
6.8 A SPECIAL CLASS OF ADAPTIVE SCALAR STEP SIZES 236
6.8.1 CONVERGENCE PROPERTIES 237
6.8.2 EXAMPLES FOR THE FUNCTION
Q
N
(R)
241
6.8.3 OPTIMAL SEQUENCE
(W
N
)
246
6.8.4 SEQUENCE
(K
N
)
247
PAR
T V TECHNICAL APPLICATION
S
7 APPROXIMATIO
N O
F TH
E PROBABILIT
Y O
F FAILURE/SURVIVA
L I
N
PLASTI
C STRUCTURA
L ANALYSI
S AND OPTIMA
L PLASTI
C DESIG
N
.
. 253
7.1 INTRODUCTION 253
7.2 PROBABILITY OF SURVIVAL/FAILURE
P
S
,PF
254
7.3 APPROXIMATION OF
P
S
,PF
BY LINEARIZATION OF TH
E TRANSFORMED
LIMIT STATE FUNCTION 257
7.3.1 THE ORIGIN LIES IN THE TRANSFORMED SAFE DOMAIN 257
7.3.2 THE ORIGIN LIES IN THE TRANSFORMED FAILURE DOMAIN .
. 260
7.4 COMPUTATION OF THE /3-POINT
Z*
262
7.5 TRUSSES 265
7.5.1 SPECIAL CASE 268
7.6 RELIABILITY-BASED DESIGN OPTIMIZATION (RBDO) 269
7.6.1 NECESSARY OPTIMALITY CONDITIONS FOR THE /?-POINT 270
7.6.2 DUALITY RELATIONS 270
PAR
T V
I APPENDI
X
A SEQUENCES
, SERIE
S AND PRODUCT
S
275
A.I MEAN VALUE THEOREMS FOR DETERMINISTIC SEQUENCES 275
A.2 ITERATIVE SOLUTION OF A LYAPUNOV MATRIX EQUATION 283
B CONVERGENC
E THEOREM
S FOR STOCHASTI
C SEQUENCE
S
287
B.I A CONVERGENCE RESULT OF ROBBINS-SIEGMUND 287
B.I.
I CONSEQUENCES 287
B.2 CONVERGENCE IN THE MEAN 290
B.3 THE STRONG LAW OF LARGE NUMBERS FOR DEPENDENT MATRIX
SEQUENCES 292
B.4 A CENTRAL LIMIT THEOREM FOR DEPENDENT VECTOR SEQUENCES.
. 293
CONTENTS XIII
C TOOL
S FROM MATRI
X CALCULU
S
295
C.I MISCELLANEOUS 295
C.2 THE V. MISES-PROCEDURE IN CASE OF ERRORS 296
REFERENCE
S
301
INDE
X 309 |
| any_adam_object | 1 |
| author | Marti, Kurt 1943- |
| author_GND | (DE-588)115845690 |
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| ctrlnum | (OCoLC)249755899 (DE-599)BVBBV019677055 |
| dewey-full | 519.62 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.62 |
| dewey-search | 519.62 |
| dewey-sort | 3519.62 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV019677055 |
| illustrated | Illustrated |
| indexdate | 2024-08-16T00:07:04Z |
| institution | BVB |
| isbn | 3540222723 9783540222729 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-013005140 |
| oclc_num | 249755899 |
| open_access_boolean | |
| owner | DE-29T DE-739 DE-20 DE-91G DE-BY-TUM DE-824 DE-1051 DE-706 DE-634 DE-11 DE-83 |
| owner_facet | DE-29T DE-739 DE-20 DE-91G DE-BY-TUM DE-824 DE-1051 DE-706 DE-634 DE-11 DE-83 |
| physical | XIII, 314 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Springer |
| record_format | marc |
| spelling | Marti, Kurt 1943- Verfasser (DE-588)115845690 aut Stochastic optimization methods Kurt Marti Berlin [u.a.] Springer 2005 XIII, 314 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 301 - 308 Auch als Internetausgabe Mathematical optimization Stochastic processes Stochastische Optimierung (DE-588)4057625-5 gnd rswk-swf Stochastische Optimierung (DE-588)4057625-5 s DE-604 http://www.gbv.de/du/services/agi/313B3B913A45FC8BC1256FF000283EDC/HM00133530 Inhaltsverzeichnis DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013005140&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marti, Kurt 1943- Stochastic optimization methods Mathematical optimization Stochastic processes Stochastische Optimierung (DE-588)4057625-5 gnd |
| subject_GND | (DE-588)4057625-5 |
| title | Stochastic optimization methods |
| title_auth | Stochastic optimization methods |
| title_exact_search | Stochastic optimization methods |
| title_full | Stochastic optimization methods Kurt Marti |
| title_fullStr | Stochastic optimization methods Kurt Marti |
| title_full_unstemmed | Stochastic optimization methods Kurt Marti |
| title_short | Stochastic optimization methods |
| title_sort | stochastic optimization methods |
| topic | Mathematical optimization Stochastic processes Stochastische Optimierung (DE-588)4057625-5 gnd |
| topic_facet | Mathematical optimization Stochastic processes Stochastische Optimierung |
| url | http://www.gbv.de/du/services/agi/313B3B913A45FC8BC1256FF000283EDC/HM00133530 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=013005140&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT martikurt stochasticoptimizationmethods |
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