## Stochastic Systems

### Solving variational inequalities with stochastic mirror-prox algorithm

#### Abstract

In this paper we consider iterative methods for stochastic variational inequalities (s.v.i.) with monotone operators. Our basic assumption is that the operator possesses both smooth and nonsmooth components. Further, only noisy observations of the problem data are available. We develop a novel Stochastic Mirror-Prox (SMP) algorithm for solving s.v.i. and show that with the convenient stepsize strategy it attains the optimal rates of convergence with respect to the problem parameters. We apply the SMP algorithm to Stochastic composite minimization and describe particular applications to Stochastic Semidefinite Feasibility problem and deterministic Eigenvalue minimization.

#### Article information

Source
Stoch. Syst. Volume 1, Number 1 (2011), 17-58.

Dates
First available in Project Euclid: 24 February 2014

http://projecteuclid.org/euclid.ssy/1393252123

Digital Object Identifier
doi:10.1214/10-SSY011

Mathematical Reviews number (MathSciNet)
MR2948917

Zentralblatt MATH identifier
1291.49006

#### Citation

Juditsky, Anatoli; Nemirovski, Arkadi; Tauvel, Claire. Solving variational inequalities with stochastic mirror-prox algorithm. Stoch. Syst. 1 (2011), no. 1, 17--58. doi:10.1214/10-SSY011. http://projecteuclid.org/euclid.ssy/1393252123.

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