On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems

Shisheng Cui, Vinayak V. Shanbhag

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Stochastic generalizations of the extragradient method are complicated by a key challenge: the scheme requires two projections on a convex set and two evaluations of the map for every major iteration. We consider two related avenues where every iteration requires a single projection: (i) A projected reflected gradient (PRG) method requiring a single evaluation of the map and a single projection; and (ii) A modified backward-forward splitting (MBFS) method that requires two evaluations of the map and a single projection. We make the following contributions: (a) We prove almost sure convergence of the iterates to a random point in the solution set for the stochastic PRG scheme under a weak sharpness requirement; (b) We prove that the mean of the gap function associated with the averaged sequence diminishes to zero at the optimal rate of O(1/√N) for both schemes where N is the iteration index.

Original languageEnglish (US)
Title of host publication2016 IEEE 55th Conference on Decision and Control, CDC 2016
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages4510-4515
Number of pages6
ISBN (Electronic)9781509018376
DOIs
StatePublished - Dec 27 2016
Event55th IEEE Conference on Decision and Control, CDC 2016 - Las Vegas, United States
Duration: Dec 12 2016Dec 14 2016

Publication series

Name2016 IEEE 55th Conference on Decision and Control, CDC 2016

Other

Other55th IEEE Conference on Decision and Control, CDC 2016
CountryUnited States
CityLas Vegas
Period12/12/1612/14/16

Fingerprint

Splitting Method
Variational Inequality Problem
Gradient Method
Monotone
Projection
Iteration
Evaluation
Gradient methods
Extragradient Method
Gap Function
Almost Sure Convergence
Optimal Rates
Sharpness
Solution Set
Iterate
Convex Sets
Gradient
Variational inequalities
Requirements
Zero

All Science Journal Classification (ASJC) codes

  • Artificial Intelligence
  • Decision Sciences (miscellaneous)
  • Control and Optimization

Cite this

Cui, S., & Shanbhag, V. V. (2016). On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems. In 2016 IEEE 55th Conference on Decision and Control, CDC 2016 (pp. 4510-4515). [7798955] (2016 IEEE 55th Conference on Decision and Control, CDC 2016). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC.2016.7798955
Cui, Shisheng ; Shanbhag, Vinayak V. / On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems. 2016 IEEE 55th Conference on Decision and Control, CDC 2016. Institute of Electrical and Electronics Engineers Inc., 2016. pp. 4510-4515 (2016 IEEE 55th Conference on Decision and Control, CDC 2016).
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Cui, S & Shanbhag, VV 2016, On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems. in 2016 IEEE 55th Conference on Decision and Control, CDC 2016., 7798955, 2016 IEEE 55th Conference on Decision and Control, CDC 2016, Institute of Electrical and Electronics Engineers Inc., pp. 4510-4515, 55th IEEE Conference on Decision and Control, CDC 2016, Las Vegas, United States, 12/12/16. https://doi.org/10.1109/CDC.2016.7798955

On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems. / Cui, Shisheng; Shanbhag, Vinayak V.

2016 IEEE 55th Conference on Decision and Control, CDC 2016. Institute of Electrical and Electronics Engineers Inc., 2016. p. 4510-4515 7798955 (2016 IEEE 55th Conference on Decision and Control, CDC 2016).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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Cui S, Shanbhag VV. On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems. In 2016 IEEE 55th Conference on Decision and Control, CDC 2016. Institute of Electrical and Electronics Engineers Inc. 2016. p. 4510-4515. 7798955. (2016 IEEE 55th Conference on Decision and Control, CDC 2016). https://doi.org/10.1109/CDC.2016.7798955