zbMATH — the first resource for mathematics

Robust stochastic approximation approach to stochastic programming. (English) Zbl 1189.90109
The authors consider the stochastic optimization problem
\[ \min_{x\in X}\left\{ f\left( x\right) =\mathbb{E}\left[ F\left( x,\xi \right) \right] \right\} , \tag{1} \] where \(X\subset \mathbb{R}^{n}\) is a nonempty bounded closed convex set, \( \xi \) is a random vector whose probability distribution \(P\) is supported on the set \(\Xi \subset \mathbb{R}^{d}\) and \(F:X\times \Xi \rightarrow \mathbb{R}\).
The aim of this paper is to compare two computational approaches for solving (1) based on Monte Carlo sampling techniques: the stochastic approximation (SA) and the sample average approximation (SAA).
The authors demonstrate that a properly modified SA approach can be competitive and even significantly outperform the SAA method for a certain class of convex stochastic problems.
They extend the analysis to the case of convex-concave stochastic saddle point problems and present results of numerical experiments. Finally, some technical proofs are given in the appendix.

90C15 Stochastic programming
90C25 Convex programming
Full Text: DOI