Variable sample-size optimistic mirror descent algorithm for stochastic mixed variational inequalities

In this paper, we propose a variable sample-size optimistic mirror descent algorithm under the Bregman distance for a class of stochastic mixed variational inequalities. Different from those conventional variable sample-size extragradient algorithms to evaluate the expected mapping twice at each iteration, our algorithm requires only one evaluation of the expected mapping and hence can significantly reduce the computation load. In the monotone case, the proposed algorithm can achieve \({\mathcal {O}}(1/t)\) ergodic convergence rate in terms of the expected restricted gap function and, under the strongly generalized monotonicity condition, the proposed algorithm has a locally linear convergence rate of the Bregman distance between iterations and solutions when the sample size increases geometrically. Furthermore, we derive some results on stochastic local stability under the generalized monotonicity condition. Numerical experiments indicate that the proposed algorithm compares favorably with some existing methods.