# Optimistic Bilevel Optimization with Composite Lower-Level Problem

arXiv: https://arxiv.org/abs/2602.05417 · HTML: https://mlsanity.com/paper/2602.05417/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Feb 2026.

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

- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

This paper introduces a novel double regularization scheme for bilevel optimization problems whose lower-level problem is composite and convex, but not necessarily strongly convex, in the lower-level variable. The analysis focuses on the primal-dual solution mapping of the regularized lower-level problem and exploits its properties to derive an almost-everywhere formula for the gradient of the regularized hyper-objective under mild assumptions. The paper then establishes conditions under which the hyper-objective of the actual problem is well defined and shows that its gradient can be approximated by the gradient of the regularized hyper-objective. Building on these results, a gradient sampling-based algorithm computes approximately stationary points of the regularized hyper-objective, and we prove its convergence to stationary points of the actual problem. Two numerical examples from machine learning demonstrate the proposed approach.