Stochastic optimization methods

Kurt Marti3540222723, 9783540222729

Optimization problems arising in practice involve random model parameters. For the computation of robust optimal solutions, i.e., optimal solutions being insensitive with respect to random parameter variations, appropriate deterministic substitute problems are needed. Based on the probability distribution of the random data, and using decision theoretical concepts, optimisation problems under stochastic uncertainty are converted into appropriate deterministic substitute problems. Due to the occurring probabilities and expectations, hence, multiple integrals, approximative solution techniques must be applied. Several deterministic and stochastic approximation methods are provided: Taylor expansion methods, regression and response surface methods (RSM), probability inequalities, First order reliability methods (FORM), convex approximation/deterministic descent directions/efficient points, stochastic gradient methods, stochastic approximation procedures, differentiation formulas for probabilities and expectations. The mathematical properties of the approximative problems and iterative solution procedures (e.g. rate of convergence) are described.

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Table of contents :
Front cover……Page 1
Title page……Page 3
Date-line……Page 4
Preface……Page 5
Contents……Page 9
Part I Basic Stochastic Optimization Methods……Page 15
1.1 Introduction……Page 17
1.2 Deterministic Substitute Problems: Basic Formulation……Page 19
1.2.1 Minimum or Bounded Expected Costs……Page 20
1.2.2 Minimum or Bounded Maximum Costs (Worst Case)……Page 22
2.1 Optimum Design Problems with Random Parameters……Page 23
2.1.1 Deterministic Substitute Problems in Optimal Design……Page 27
2.1.2 Deterministic Substitute Problems in Quality Engineering……Page 30
2.2 Basic Properties of Substitute Problems……Page 32
2.3.1 Approximation of the Loss Function……Page 34
2.3.2 Regression Techniques, RSM……Page 36
2.3.3 Taylor Expansion Methods……Page 39
2.4 Applications to Problems in Quality Engineering……Page 42
2.5 Approximation of Probabilities – Probability Inequalities……Page 43
2.5.1 Bonferroni-Type Inequalities……Page 44
2.5.2 Tschebyscheff-Type Inequalities……Page 45
2.5.3 FORM (First Order Reliability Methods)……Page 50
2.6.1 Plastic Analysis and Optimal Plastic Design……Page 52
2.6.2 Optimal Elastic Design……Page 55
Part II Differentiation Methods……Page 57
3.1 Introduction……Page 59
3.2 Transformation Method: Differentiation by Using an Integral Transformation……Page 62
3.2.1 Representation of the Derivatives by Surface Integrals……Page 67
3.3 The Differentiation of Structural Reliabilities……Page 70
3.4.1 More General Response (State) Functions……Page 72
3.5.1 Computation of Structural Reliabilities and its Sensitivities……Page 77
3.5.2 Numerical Computation of Derivatives of the Probability Functions Arising in Chance Constrained Programming……Page 81
3.6 Integral Representations of the Probability Function $P(x)$ and its Derivatives……Page 87
3.7 Orthogonal Function Series Expansions I: Expansions in Hermite Functions, Case $m=1$……Page 90
3.7.1 Integrals over the Basis Functions and the Coefficients of the Orthogonal Series……Page 94
3.7.2 Estimation/Approximation of $P(x)$ and its Derivatives……Page 97
3.7.3 The Integrated Square Error (ISE) of Deterministic Approximations……Page 103
3.8 Orthogonal Function Series Expansions II: Expansions in Hermite Functions, Case $m>1$……Page 104
3.9.1 Expansions in Trigonometric and Legendre Series……Page 106
3.9.2 Expansions in Laguerre Series……Page 107
Part III Deterministic Descent Directions……Page 109
4.1 Convex Approximation……Page 111
4.1.1 Approximative Convex Optimization Problem……Page 115
4.2 Computation of Descent Directions in Case of Normal Distributions……Page 117
4.2.1 Descent Directions of Convex Programs……Page 121
4.2.2 Solution of the Auxiliary Programs……Page 124
4.3 Efficient Solutions (Points)……Page 129
4.3.1 Necessary Optimality Conditions Without Gradients……Page 131
4.4 Descent Directions in Case of Elliptically Contoured Distributions……Page 133
4.5 Construction of Descent Directions by Using Quadratic Approximations of the Loss Function……Page 136
Part IV Semi-Stochastic Approximation Methods……Page 141
5.1 Introduction……Page 143
5.2 Gradient Estimation Using the Response Surface Methodology (RSM)……Page 145
5.2.1 The Two Phases of RSM……Page 148
5.2.2 The Mean Square Error of the Gradient Estimator……Page 152
5.3 Estimation of the Mean Square (Mean Functional) Error……Page 156
5.3.1 The Argument Case……Page 157
5.4.1 Asymptotically Correct Response Surface Model……Page 161
5.4.2 Biased Response Surface Model……Page 164
5.5 Convergence Rates of Hybrid Stochastic Approximation Procedures……Page 167
5.5.1 Fixed Rate of Stochastic and Deterministic Steps……Page 172
5.5.2 Lower Bounds for the Mean Square Error……Page 182
5.5.3 Decreasing Rate of Stochastic Steps……Page 186
6.1 Introduction……Page 191
6.2 Solution of Optimality Conditions……Page 192
6.3 General Assumptions and Notations……Page 193
6.3.1 Interpretation of the Assumptions……Page 195
6.3.2 Notations and Abbreviations in this Chapter……Page 196
6.4.1 Estimation of the Quadratic Error……Page 197
6.4.2 Consideration of the Weighted Error Sequence……Page 199
6.4.3 Further Preliminary Results……Page 202
6.5.1 Convergence with Probability One……Page 204
6.5.2 Convergence in the Mean……Page 206
6.5.3 Convergence in Distribution……Page 209
6.6 Realisation of Search Directions $Y_n$……Page 218
6.6.1 Estimation of $G^ast$……Page 222
6.6.2 Update of the Jacobian……Page 223
6.6.3 Estimation of Error Variances……Page 228
6.7 Realization of Adaptive Step Sizes……Page 233
6.7.1 Optimal Matrix Step Sizes……Page 234
6.7.2 Adaptive Scalar Step Size……Page 240
6.8 A Special Class of Adaptive Scalar Step Sizes……Page 250
6.8.1 Convergence Properties……Page 251
6.8.2 Examples for the Function $Q_n(r)$……Page 255
6.8.3 Optimal Sequence ($w_n$)……Page 260
6.8.4 Sequence ($K_n$)……Page 261
Part V Technical Applications……Page 265
7.1 Introduction……Page 267
7.2 Probability of Survival/Failure $p_s$,$p_f$……Page 268
7.3.1 The Origin Lies in the Transformed Safe Domain……Page 271
7.3.2 The Origin Lies in the Transformed Failure Domain……Page 274
7.4 Computation of the $beta$-Point $z^ast$……Page 276
7.5 Trusses……Page 279
7.5.1 Special Case……Page 282
7.6 Reliability-Based Design Optimization (RBDO)……Page 283
7.6.2 Duality Relations……Page 284
Part VI Appendix……Page 287
A.1 Mean Value Theorems for Deterministic Sequences……Page 289
A.2 Iterative Solution of a Lyapunov Matrix Equation……Page 297
B.1.1 Consequences……Page 301
B.2 Convergence in the Mean……Page 304
B.3 The Strong Law of Large Numbers for Dependent Matrix Sequences……Page 306
B.4 A Central Limit Theorem for Dependent Vector Sequences……Page 307
C.1 Miscellaneous……Page 309
C.2 The v. Mises-Procedure in Case of Errors……Page 310
References……Page 315
Index……Page 323
Back cover……Page 329