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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.

##### MSC:
 90C15 Stochastic programming 90C25 Convex programming
##### Keywords:
Monte Carlo sampling; minimax problems; saddle point
SUTIL
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