Stochastic Systems

Solving variational inequalities with stochastic mirror-prox algorithm

Anatoli Juditsky, Arkadi Nemirovski, and Claire Tauvel

Full-text: Open access


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

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

First available in Project Euclid: 24 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90C15: Stochastic programming 65K10: Optimization and variational techniques [See also 49Mxx, 93B40]
Secondary: 90C47: Minimax problems [See also 49K35]

Variational inequalities with monotone operators stochastic convex-concave saddle-point problem large scale stochastic approximation reduced complexity algorithms for convex optimization


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.

Export citation


  • [1] Azuma, K. (1967). Weighted sums of certain dependent random variables. Tökuku Math. J. 19, 357–367.
  • [2] Ben-Tal, A., Nemirovski, A. (2005). Non-Euclidean restricted memory level method for large-scale convex optimization. Math. Progr. 102, 407–456.
  • [3] Grigoriadis, M.D., Khachiyan, L.G. (1995). A sublinear-time randomized approximation algorithm for matrix games. OR Letters 18, 53–58.
  • [4] Juditsky, A., Lan, G., Nemirovski, A., Shapiro, A. (2009) Stochastic Approximation approach to Stochastic Programming. SIAM J. Optim. 19, 1574–1609.
  • [5] Juditsky, A., Nemirovski, A. (2008). Large deviations of vector-valued martingales in 2-smooth normed spaces E-print:
  • [6] Juditsky, A., Kilinç Karzan, F., Nemirovski, A. (2010). $L_{1}$ minimization via randomized first order algorithms. Submitted to Mathematical Programming E-print:
  • [7] Korpelevich, G. (1983). Extrapolation gradient methods and relation to Modified Lagrangeans” Ekonomika i Matematicheskie Metody 19, 694–703 (in Russian; English translation in Matekon).
  • [8] Nemirovski, A., Yudin, D. (1983). Problem complexity and method efficiency in Optimization, J. Wiley & Sons.
  • [9] Nemirovski, A. (2004). Prox-method with rate of convergence $O(1/t)$ for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIAM J. Optim. 15, 229–251.
  • [10] Lu, Z., Nemirovski A., Monteiro, R. (2007). Large-scale Semidefinite Programming via saddle point Mirror-Prox algorithm. Math. Progr. 109, 211–237.
  • [11] Nemirovski, A., Onn, S., Rothblum, U. (2010). Accuracy certificates for computational problems with convex structure. Math. of Oper. Res. 35:1, 52–78.
  • [12] Nesterov, Yu. (2005). Smooth minimization of non-smooth functions. Math. Progr. 103, 127–152.
  • [13] Nesterov, Yu. (2005). Excessive gap technique in nonsmooth convex minimization. SIAM J. Optim. 16, 235–249.
  • [14] Nesterov, Yu. (2007). Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Progr. 109, 319–344.
  • [15] Rubinfeld, R. (2006). Sublinear time algorithms. In: Marta Sanz-Solé, Javier Soria, Juan Luis Varona, Joan Verdera, Eds. International Congress of Mathematicians, Madrid 2006, Vol. III, 1095–11110. European Mathematical Society Publishing House.