Stochastic Optimization Methods (eBook)

Preface 5
Contents 9
Part I Basic Stochastic Optimization Methods 15
1 Decision/Control Under Stochastic Uncertainty 16
1.1 Introduction 16
1.2 Deterministic Substitute Problems: Basic Formulation 18
2 Deterministic Substitute Problems in Optimal Decision Under Stochastic Uncertainty 22
2.1 Optimum Design Problems with Random Parameters 22
2.2 Basic Properties of Substitute Problems 31
2.3 Approximations of Deterministic Substitute Problems in Optimal Design 33
2.4 Applications to Problems in Quality Engineering 41
2.5 Approximation of Probabilities - Probability Inequalities 42
2.6 Construction of State Functions in Structural Analysis and Design 51
Part II Differentiation Methods 57
3 Differentiation Methods for Probability and Risk Functions 58
3.1 Introduction 58
3.2 Transformation Method: Differentiation by Using an Integral Transformation 61
3.3 The Differentiation of Structural Reliabilities 69
3.4 Extensions 71
3.5 Computation of Probabilities and its Derivatives by Asymptotic Expansions of Integral of Laplace Type 76
3.6 Integral Representations of the Probability Function P(x) and its Derivatives 86
3.7 Orthogonal Function Series Expansions I: Expansions in Hermite Functions, Case m = 1 89
3.8 Orthogonal Function Series 103
3.9 Orthogonal Function Series Expansions III: Expansions in Trigonometric, Legendre and Laguerre Series 105
Part III Deterministic Descent Directions 108
4 Deterministic Descent Directions and Efficient Points 110
4.1 Convex Approximation 110
4.2 Computation of Descent Directions in Case of Normal Distributions 116
4.3 Efficient Solutions ( Points) 128
4.4 Descent Directions in Case of Elliptically Contoured Distributions 132
4.5 Construction of Descent Directions by Using Quadratic Approximations of the Loss Function 135
Part IV Semi-Stochastic Approximation Methods 141
5 RSM-Based Stochastic Gradient Procedures 142
5.1 Introduction 142
5.2 Gradient Estimation Using the Response Surface Methodology (RSM) 144
5.3 Estimation of the Mean Square (Mean Functional) Error 155
5.4 Convergence Behavior of Hybrid Stochastic Approximation Methods 160
5.5 Convergence Rates of Hybrid Stochastic Approximation Procedures 166
6 Stochastic Approximation Methods with Changing Error Variances 190
6.1 Introduction 190
6.2 Solution of Optimality Conditions 191
6.3 General Assumptions and Notations 192
6.4 Preliminary Results 196
6.5 General Convergence Results 203
6.6 Realisation of Search Directions 217
6.7 Realization of Adaptive Step Sizes 232
6.8 A Special Class of Adaptive Scalar Step Sizes 249
Part V Technical Applications 264
7 Approximation of the Probability of Failure/Survival in Plastic Structural Analysis and Optimal Plastic Design 266
7.1 Introduction 266
7.2 Probability of Survival/Failure p8,Pf 267
7.3 Approximation of ps,Pf by Linearization of the Transformed Limit State Function 270
7.4 Computation of the ß-Point z 275
7.5 Trusses 278
7.6 Reliability- Based Design Optimization ( RBDO) 282
Part VI Appendix 286
A Sequences, Series and Products 288
A.1 Mean Value Theorems for Deterministic Sequences 288
A.2 Iterative Solution of a Lyapunov Matrix Equation 296
B Convergence Theorems for Stochastic Sequences 300
B.1 A Convergence Result of Robbins-Siegmund 300
B.2 Convergence in the Mean 303
B.3 The Strong Law of Large Numbers for Dependent Matrix Sequences 305
B.4 A Central Limit Theorem for Dependent Vector Sequences 306
C Tools from Matrix Calculus 308
C.1 Miscellaneous 308
C. 2 The v. Mises- Procedure in Case of Errors 309
Index 322