A Block-Activated Decomposition Algorithm for Multi-Stage Stochastic Variational Inequalities
Minh N. Bùi
TL;DR
The paper addresses solving a multi-stage stochastic variational inequality with nonanticipativity constraints by formulating it as finding $x \in \mathcal{V}$ and $v^* \in \mathcal{V}^{\perp}$ such that $-v^*(\xi) \in A(\xi,x(\xi)) + N_{C(\xi)}(x(\xi))$ for all $\xi \in \Xi$. It introduces a block-activated decomposition algorithm based on projective splitting that updates only a subset of scenarios per iteration and treats the resolvent $J_{\gamma A}$ and the projection onto $C$ separately, improving computational tractability over methods relying on the sum of operators. The method is proven convergent to a solution $(\overline{x},\overline{v}^*)$ with $x_n \to \overline{x}$ and $v_n^* \to \overline{v}^*$ and supports asynchronous updates. In CVaR-based risk-averse stochastic programming, the framework yields a reduction to projections and a univariate proximal equation, avoiding complex constrained subproblem solves and enabling scalable solutions. Overall, the approach fills a gap between block-activated operator splitting and variational inequality formulations, offering practical benefits for large-scale multi-stage problems.
Abstract
We develop a block-activated decomposition algorithm for multi-stage stochastic variational inequalities with nonanticipativity constraints, which offers two computational novelties: (i) At each iteration, our method activates only a user-chosen block of scenarios. (ii) For each activated scenario, it employs the resolvent of the cost operator and the projector onto the constraint set separately. These features enhance computational 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 objectives, our method requires only the projections onto constraint sets, together with solving a univariate equation involving the proximity operators of the cost functions.