# Set Smoothness Unlocks Clarke Hyper-stationarity in Bilevel Optimization

arXiv: https://arxiv.org/abs/2506.04587 · HTML: https://mlsanity.com/paper/2506.04587/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2025.

## Built on by (most cited first, 5 of 5)

- [Select-then-differentiate: Solving Bilevel Optimization with Manifold Lower-level Solution Sets](https://mlsanity.com/paper/2605.09209/) (2026)
- [Faster Gradient Methods for Highly-Smooth Stochastic Bilevel Optimization](https://mlsanity.com/paper/2509.02937/) (2025)
- [Nonconvex Decentralized Stochastic Bilevel Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2509.15543/) (2025)
- [Bilevel Models for Adversarial Learning and A Case Study](https://mlsanity.com/paper/2510.25121/) (2025)
- [Zeroth-Order Stackelberg Control in Combinatorial Congestion Games](https://mlsanity.com/paper/2602.23277/) (2026)

## Abstract (arXiv)

Solving bilevel optimization (BLO) problems to global optimality is generally intractable. A common surrogate is to compute a hyper-stationary point -- a stationary point of the hyper-objective function obtained by minimizing or maximizing the upper-level objective over the lower-level solution set. Existing methods, however, either provide weak notions of stationarity or require restrictive assumptions to guarantee the smoothness of hyper-objective functions. In this paper, we eliminate these impractical assumptions and show that strong (Clarke) hyper-stationarity remains computable even when the hyper-objective is nonsmooth. Our key ingredient is a new structural property, called set smoothness, which captures the variational dependence of the lower-level solution set on the upper-level variable. We prove that this property holds for a broad class of BLO problems and ensures weak convexity (resp. concavity) of pessimistic (resp. optimistic) hyper-objective functions. Building on this foundation, we show that a zeroth-order algorithm that computes approximate Clarke hyper-stationary points with non-asymptotic convergence guarantees. To the best of our knowledge, this is the first computational guarantee for Clarke-type stationarity in nonsmooth BLO. Beyond this specific application, the set smoothness property emerges as a structural concept of independent interest, with potential to inform the analysis of broader classes of optimization and variational problems.