# Alternating Gradient-Type Algorithm for Bilevel Optimization with Inexact Lower-Level Solutions via Moreau Envelope-based Reformulation

arXiv: https://arxiv.org/abs/2412.18929 · HTML: https://mlsanity.com/paper/2412.18929/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Dec 2024.

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

- [Co-adaptive Rule Routing and Constraining for Structured Knowledge Retrieval](https://mlsanity.com/paper/2610.14465/) (2026)

## Abstract (arXiv)

In this paper, we study a class of bilevel optimization problems where the lower-level problem is a convex composite optimization model, which arises in various applications, including bilevel hyperparameter selection for regularized regression models. To solve these problems, we propose an Alternating Gradient-type algorithm with Inexact Lower-level Solutions (AGILS) based on a Moreau envelope-based reformulation of the bilevel optimization problem. The proposed algorithm does not require exact solutions of the lower-level problem at each iteration, improving computational efficiency. We prove the convergence of AGILS to stationary points and, under the Kurdyka-Lojasiewicz (KL) property, establish its sequential convergence. Numerical experiments, including a toy example and a bilevel hyperparameter selection problem for the sparse group Lasso model, demonstrate the effectiveness of the proposed AGILS.