Abstract
The paper deals with stochastic mathematical programming problems with additional difficulty brought up by the presence of infinitely many constraints. A sample average approximation (SAA) approach is proposed and properties of SAA estimators, analyzed. It is shown that for the case of deterministic constraints, SAA estimators are consistent and meet requirements to qualify the estimates as acceptable approximations of unknown values. Moreover, good error bounds of the estimate of the objective function are provided. We also discuss a Galerkin-like scheme and a tailored cutting-plane method for solving resulting semi-infinite SAA problems. The framework is then extended to the case of random constraints through Karhunen–Loève expansion or constraints approximation via sampling. A general procedure for solving stochastic semi-infinite programming problems, that lends itself better for parallel computation, is then described. Numerical examples are also included for the sake of illustration.
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