### 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 language | English (US) |
---|---|

Pages (from-to) | 789-819 |

Number of pages | 31 |

Journal | Set-Valued and Variational Analysis |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2018 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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*Set-Valued and Variational Analysis*, vol. 26, no. 4, pp. 789-819. https://doi.org/10.1007/s11228-018-0472-9

**On Stochastic Mirror-prox Algorithms for Stochastic Cartesian Variational Inequalities : Randomized Block Coordinate and Optimal Averaging Schemes.** / Yousefian, Farzad; Nedić, Angelia; Shanbhag, Vinayak V.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On Stochastic Mirror-prox Algorithms for Stochastic Cartesian Variational Inequalities

T2 - Randomized Block Coordinate and Optimal Averaging Schemes

AU - Yousefian, Farzad

AU - Nedić, Angelia

AU - Shanbhag, Vinayak V.

PY - 2018/12/1

Y1 - 2018/12/1

N2 - 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.

AB - 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.

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U2 - 10.1007/s11228-018-0472-9

DO - 10.1007/s11228-018-0472-9

M3 - Article

AN - SCOPUS:85057805789

VL - 26

SP - 789

EP - 819

JO - Set-Valued and Variational Analysis

JF - Set-Valued and Variational Analysis

SN - 1877-0533

IS - 4

ER -