# Regularization methods for solving hierarchical variational inequalities with complexity guarantees

arXiv: https://arxiv.org/abs/2512.20772 · HTML: https://mlsanity.com/paper/2512.20772/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2025.

## Built on by (most cited first, 1 of 1)

- [Stochastic Nonconvex Bilevel Optimization: Improved Rates Without Rare-Visit Assumption](https://mlsanity.com/paper/2609.06580/) (2026)

## Abstract (arXiv)

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.