A prediction-correction ADMM for multistage stochastic variational inequalities

The multistage stochastic variational inequality is reformulated into a variational inequality with separable structure through introducing a new variable. The prediction-correction ADMM which was originally proposed in [B.-S. He, L.-Z. Liao and M.-J. Qian, J. Comput. Math., 24 (2006), 693--710] for solving deterministic variational inequalities in finite dimensional spaces is adapted to solve the multistage stochastic variational inequality. Weak convergence of the sequence generated by that algorithm is proved under the conditions of monotonicity and Lipschitz continuity. When the sample space is a finite set, the corresponding multistage stochastic variational inequality is defined on a finite dimensional Hilbert space and the strong convergence of the sequence naturally holds true. A numerical example in that case is given to show the efficiency of the algorithm.