# Theoretical smoothing frameworks for nonsmooth simple bilevel problems

arXiv: https://arxiv.org/abs/2401.17852 · HTML: https://mlsanity.com/paper/2401.17852/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Jan 2024.

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

- [A Framework for Bilevel Optimization on Riemannian Manifolds](https://mlsanity.com/paper/2402.03883/) (2024)

## Abstract (arXiv)

Bilevel programming has recently received a great deal of attention due to its abundant applications in many areas. The optimal value function approach provides a useful reformulation of the bilevel problem, but its utility is often limited due to the nonsmoothness of the value function even in cases when the associated lower-level function is smooth. In this paper, we present two smoothing strategies for the value function associated with lower-level functions that are not necessarily smooth but are Lipschitz continuous. The first method employs quadratic regularization for partially convex lower-level functions, while the second utilizes entropic regularization for general lower-level objective functions. Meanwhile, the property known as gradient consistency is crucial in ensuring that a designed smoothing algorithm is globally subsequentially convergent to stationary points of the value function reformulation. With this motivation, we prove that the proposed smooth approximations satisfy the gradient consistent property under certain conditions on the lower-level function.