On Stochastic Mirror-prox Algorithms for Stochastic Cartesian Variational Inequalities: Randomized Block Coordinate and Optimal Averaging Schemes

Farzad Yousefian, Angelia Nedić, Vinayak V. Shanbhag

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Motivated by multi-user optimization problems and non-cooperative Nash games in uncertain regimes, we consider stochastic Cartesian variational inequality problems where the set is given as the Cartesian product of a collection of component sets. First, we consider the case where the number of the component sets is large and develop a randomized block stochastic mirror-prox algorithm, where at each iteration only a randomly selected block coordinate of the solution vector is updated through implementing two consecutive projection steps. We show that when the mapping is strictly pseudo-monotone, the algorithm generates a sequence of iterates that converges to the solution of the problem almost surely. When the maps are strongly pseudo-monotone, we prove that the mean-squared error diminishes at the optimal rate. Second, we consider large-scale stochastic optimization problems with convex objectives and develop a new averaging scheme for the randomized block stochastic mirror-prox algorithm. We show that by using a different set of weights than those employed in the classical stochastic mirror-prox methods, the objective values of the averaged sequence converges to the optimal value in the mean sense at an optimal rate. Third, we consider stochastic Cartesian variational inequality problems and develop a stochastic mirror-prox algorithm that employs the new weighted averaging scheme. We show that the expected value of a suitably defined gap function converges to zero at an optimal rate.

Original languageEnglish (US)
Pages (from-to)789-819
Number of pages31
JournalSet-Valued and Variational Analysis
Volume26
Issue number4
DOIs
StatePublished - Dec 1 2018

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Cartesian
Variational Inequalities
Averaging
Mirror
Mirrors
Optimal Rates
Pseudomonotone
Variational Inequality Problem
Converge
Optimization Problem
Gap Function
Large-scale Optimization
Stochastic Optimization
Cartesian product
Expected Value
Mean Squared Error
Iterate
Consecutive
Strictly
Projection

All Science Journal Classification (ASJC) codes

  • Analysis
  • Statistics and Probability
  • Numerical Analysis
  • Geometry and Topology
  • Applied Mathematics

Cite this

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