On the solution of monotone nested variational inequalities

TL;DR

This paper introduces a new algorithm for solving nested variational inequalities under minimal assumptions, providing the first complexity analysis for such problems and ensuring convergence with weak conditions.

Contribution

It proposes a projected averaging Tikhonov algorithm for nested variational inequalities requiring only monotonicity, and offers the first complexity analysis considering optimality at both levels.

Findings

Algorithm guarantees convergence under weak monotonicity conditions.

Provides the first complexity analysis for nested variational inequalities.

Achieves solutions with minimal assumptions on problem structure.

Abstract

We study nested variational inequalities, which are variational inequalities whose feasible set is the solution set of another variational inequality. We present a projected averaging Tikhonov algorithm requiring the weakest conditions in the literature to guarantee the convergence to solutions of the nested variational inequality. Specifically, we only need monotonicity of the upper- and the lower-level variational inequalities. Also, we provide the first complexity analysis for nested variational inequalities considering optimality of both the upper- and lower-level.