A Block-Activated Decomposition Algorithm for Multi-Stage Stochastic Variational Inequalities

We develop a block-activated decomposition algorithm for multi-stage stochastic variational inequalities with nonanticipativity constraints, which features two computational novelties: (i) At each iteration, our method activates only a user-chosen subset of scenarios. (ii) For each activated scenario, it employs the resolvent of the cost operator and the projector onto the constraint set separately. These reduce computational load and enhance tractability, in contrast with existing approaches, which often rely on evaluating the resolvent of the sum of the cost operator and normal cone operator of the constraint set. As an application, we demonstrate that in risk-averse stochastic programming with conditional value-at-risk objective functions, our method requires only projecting onto constraint sets, together with solving univariate equations involving the proximity operators of the cost functions, thereby avoiding solving high-dimensional constrained subproblems as required by existing methods.