On Multistage Pseudomonotone Stochastic Variational Inequalities

This paper focuses on the solvability of multistage pseudomonotone stochastic variational inequalities (SVIs). On the one hand, some known solvability results of pseudomonotone deterministic variational inequalities cannot be directly extended to multistage pseudomonotone SVIs, so we construct the isomorphism between both and then establish theoretical results on the existence, convexity, boundedness and compactness of the solution set for multistage pseudomonotone SVIs via such an isomorphism. On the other hand, there does not exist a special algorithm for solving the multistage pseudomonotone SVIs so far, so we propose some sufficient conditions on the elicitability of multistage pseudomonotone SVIs, which opens the door for applying Rockafellar’s elicited progressive hedging algorithm to solve such SVIs. Numerical results on solving a two-stage stochastic market optimization problem and randomly generated two-stage pseudomonotone linear complementarity problems are presented.