Minibatch Forward-Backward-Forward Methods for Solving Stochastic Variational Inequalities

We develop a new stochastic algorithm for solving pseudomonotone stochastic variational inequalities. Our method builds on Tseng’s forward-backward-forward algorithm, which is known in the deterministic literature to be a valuable alternative to Korpelevich’s extragradient method when solving variational inequalities over a convex and closed set governed by pseudomonotone Lipschitz continuous operators. The main computational advantage of Tseng’s algorithm is that it relies only on a single projection step and two independent queries of a stochastic oracle. Our algorithm incorporates a minibatch sampling mechanism and leads to almost sure convergence to an optimal solution. To the best of our knowledge, this is the first stochastic look-ahead algorithm achieving this by using only a single projection at each iteration.

A Model of Multistage Risk-Averse Stochastic Optimization and its Solution by Scenario-Based Decomposition Algorithms

Stochastic optimization models based on risk-averse measures are of essential importance in financial management and business operations. This paper studies new algorithms for a popular class of these models, namely, the mean-deviation models in multistage decision making under uncertainty. It is argued that these types of problems enjoy a scenario-decomposable structure, which could be utilized in an efficient progressive hedging procedure. In case that linkage constraints arise in reformulations of the original problem, a Lagrange progressive hedging algorithm could be utilized to solve the reformulated problem. Convergence results of the algorithms are obtained based on the recent development of the Lagrangian form of stochastic variational inequalities. Numerical results are provided to show the effectiveness of the proposed algorithms.

On a class of generalized stochastic Browder mixed variational inequalities

In this paper, we introduce a class of stochastic variational inequalities generated from the Browder variational inequalities. First, the existence of solutions for these generalized stochastic Browder mixed variational inequalities (GS-BMVI) are investigated based on FKKM theorem and Aummann’s measurable selection theorem. Then the uniqueness of solution for GS-BMVI is proved based on strengthening conditions of monotonicity and convexity, the compactness and convexity of the solution sets are discussed by Minty’s technique. The results of this paper can provide a foundation for further research of a class of stochastic evolutionary problems driven by GS-BMVI.

Two-Stage Stochastic Variational Inequalities: Theory, Algorithms and Applications

The stochastic variational inequality (SVI) provides a unified form of optimality conditions of stochastic optimization and stochastic games which have wide applications in science, engineering, economics and finance. In the recent two decades, one-stage SVI has been studied extensively and widely used in modeling equilibrium problems under uncertainty. Moreover, the recently proposed two-stage SVI and multistage SVI can be applied to the case when the decision makers want to make decisions at different stages in a stochastic environment. The two-stage SVI is a foundation of multistage SVI, which is to find a pair of “here-and-now” solution and “wait-and-see” solution. This paper provides a survey of recent developments in analysis, algorithms and applications of the two-stage SVI.