Regularization methods for solving hierarchical variational inequalities with complexity guarantees

TL;DR

This paper introduces a novel regularization-based double-loop method for hierarchical variational inequalities, providing convergence guarantees and rate analysis, applicable to a broad class of structured problems.

Contribution

It develops a flexible regularization and penalization approach with proven convergence and rates for hierarchical variational inequalities in Hilbert spaces.

Findings

Method converges asymptotically with proven rates

Applicable to structured operator splitting problems

Validated through numerical experiments

Abstract

We consider hierarchical variational inequality problems, or more generally, variational inequalities defined over the set of zeros of a monotone operator. This framework includes convex optimization over equilibrium constraints and equilibrium selection problems. In a real Hilbert space setting, we combine a Tikhonov regularization and a proximal penalization to develop a flexible double-loop method for which we prove asymptotic convergence and provide rate statements in terms of gap functions. Our method is flexible, and effectively accommodates a large class of structured operator splitting formulations for which fixed-point encodings are available. Finally, we validate our findings numerically on various examples.