Solving stochastic structural optimization problems by RSM-based stochastic approximation methods — gradient estimation in case of intermediate variables

Reliability-based structural optimization methods use mostly the following basic design criteria: I) Minimum weight (volume or costs) and II) high strength of the structure. Since several parameters of the structure, e.g. material parameters, loads, manufacturing errors, are not given, fixed quantities, but random variables having a certain probability distribution P,stochastic optimization problems result from criteria (I), (II), which can be represented by(1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathop {\min }\limits_{x \in D} F(x)withF(x): = Ef(\omega ,x).$$ \end{document} Here,f=f(ω,x) is a function on ℛr depending on a random element ω, “E” denotes the expectation operator andD is a given closed, convex subset of ℛr. Stochastic approximation methods are considered for solving (1), where gradient estimators are obtained by means of the response surface methodology (RSM). Moreover, improvements of the RSM-gradient estimator by using “intermediate” or “intervening” variables are examined.