Set-Valued and Variational Analysis

, Volume 26, Issue 4, pp 789–819 | Cite as

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

  • Farzad YousefianEmail author
  • Angelia Nedić
  • Uday V. Shanbhag


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.


Stochastic variational inequalities Nash games Randomized block coordinate methods Stochastic mirror-prox methods Stochastic approximation methods 

Mathematics Subject Classification (2010)

65K10 90C33 90C06 91A10 


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Farzad Yousefian
    • 1
    Email author
  • Angelia Nedić
    • 2
  • Uday V. Shanbhag
    • 3
  1. 1.School of Industrial Engineering and ManagementOklahoma State UniversityStillwaterUSA
  2. 2.School of Electrical, Computer and Energy EngineeringArizona State UniversityTempeUSA
  3. 3.Department of Industrial and Manufacturing EngineeringPennsylvania State UniversityUniversity ParkUSA

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