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Computational Optimization and Applications

, Volume 74, Issue 3, pp 779–820 | Cite as

Optimal stochastic extragradient schemes for pseudomonotone stochastic variational inequality problems and their variants

  • Aswin Kannan
  • Uday V. ShanbhagEmail author
Article

Abstract

We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to monotone maps. However, non-monotone stochastic variational inequality problems are not uncommon and are seen to arise from product pricing, fractional optimization problems, and subclasses of economic equilibrium problems. Motivated by the need to address a broader class of maps, we make the following contributions: (1) we present an extragradient-based stochastic approximation scheme and prove that the iterates converge to a solution of the original problem under either pseudomonotonicity requirements or a suitably defined acute angle condition. Such statements are shown to be generalizable to the stochastic mirror-prox framework; (2) under strong pseudomonotonicity, we show that the mean-squared error in the solution iterates produced by the extragradient SA scheme converges at the optimal rate of \({{\mathcal {O}}}\left( \frac{1}{{K}}\right) \), statements that were hitherto unavailable in this regime. Notably, we optimize the initial steplength by obtaining an \(\epsilon \)-infimum of a discontinuous nonconvex function. Similar statements are derived for mirror-prox generalizations and can accommodate monotone SVIs under a weak-sharpness requirement. Finally, both the asymptotics and the empirical rates of the schemes are studied on a set of variational problems where it is seen that the theoretically specified initial steplength leads to significant performance benefits.

Keywords

Variational inequality problems Stochastic approximation Pseudomonotone 

Notes

Acknowledgements

The authors are grateful to Dr. Farzad Yousefian for his valuable suggestions on a previous version. We particularly appreciate the comments of the referees and the editor, all of which have led to significant improvements in the manuscript.

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Copyright information

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Authors and Affiliations

  1. 1.India Research Laboratory (IRL), IBM ResearchChennaiIndia
  2. 2.Industrial and Manufacturing EngineeringThe Pennsylvania State UniversityUniversity ParkUSA

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