### 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 language | English (US) |
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Title of host publication | 2016 IEEE 55th Conference on Decision and Control, CDC 2016 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 4510-4515 |

Number of pages | 6 |

ISBN (Electronic) | 9781509018376 |

DOIs | |

State | Published - Dec 27 2016 |

Event | 55th IEEE Conference on Decision and Control, CDC 2016 - Las Vegas, United States Duration: Dec 12 2016 → Dec 14 2016 |

### Publication series

Name | 2016 IEEE 55th Conference on Decision and Control, CDC 2016 |
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### Other

Other | 55th IEEE Conference on Decision and Control, CDC 2016 |
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Country | United States |

City | Las Vegas |

Period | 12/12/16 → 12/14/16 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*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

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*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - On the analysis of reflected gradient and splitting methods for monotone stochastic variational inequality problems

AU - Cui, Shisheng

AU - Shanbhag, Vinayak V.

PY - 2016/12/27

Y1 - 2016/12/27

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85010767558&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85010767558&partnerID=8YFLogxK

U2 - 10.1109/CDC.2016.7798955

DO - 10.1109/CDC.2016.7798955

M3 - Conference contribution

T3 - 2016 IEEE 55th Conference on Decision and Control, CDC 2016

SP - 4510

EP - 4515

BT - 2016 IEEE 55th Conference on Decision and Control, CDC 2016

PB - Institute of Electrical and Electronics Engineers Inc.

ER -