# Tikhonov regularized exterior penalty methods for hierarchical variational inequalities

## TL;DR

This paper introduces a novel double loop prox-penalization algorithm for solving hierarchical variational inequalities, with strong convergence guarantees and applications to bilevel optimization and multi-follower games.

## Contribution

It develops a new algorithm for nested variational inequalities with proven convergence, expanding solution methods for hierarchical equilibrium problems.

## Key findings

- Algorithm demonstrates strong convergence in Hilbert spaces.
- Applicable to hierarchical convex bilevel problems and multi-follower games.
- Preliminary numerical results show promising performance.

## Abstract

We consider nested variational inequalities con- sisting in a (upper-level) variational inequality whose feasible set is given by the solution set of another (lower-level) variational inequality. This class of hierarchical equilibrium contains a wealth of important applications, including purely hierarchical convex bilevel optimization problems and certain multi-follower games. Working within a real Hilbert space setting, we develop a double loop prox-penalization algorithm with strong conver- gence guarantees towards a solution of the nested VI problem. We present various application that fit into our framework and present also some preliminary numerical results.