A prediction-correction ADMM for multistage stochastic variational inequalities

TL;DR

This paper adapts a prediction-correction ADMM algorithm to solve multistage stochastic variational inequalities, proving convergence under certain conditions and demonstrating efficiency through a numerical example.

Contribution

It extends the prediction-correction ADMM to multistage stochastic variational inequalities with convergence analysis and practical validation.

Findings

Weak convergence under monotonicity and Lipschitz conditions

Strong convergence when the sample space is finite

Numerical example confirms algorithm efficiency

Abstract

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.