Set Smoothness Unlocks Clarke Hyper-stationarity in Bilevel Optimization
He Chen, Jiajin Li, Anthony Man-Cho So
Solving bilevel optimization (BLO) problems to global optimality is generally intractable. A common alternative is to compute a hyper-stationary point—a stationary point of the hyper-objective function formed by minimizing/maximizing the upper-level function over the lower-level solution set. However, existing approaches either yield weak notions of stationarity or rely on restrictive assumptions to ensure the smoothness of hyper-objective functions. In this paper, we remove these impractical assumptions and show that *strong* (Clarke) hyper-stationarity is still computable even when the hyper-objective is nonsmooth. Our key tool is a new structural condition, called *set smoothness*, which captures the variational relationship between the lower-level solution set and the upper-level variable. We prove that this condition holds for a broad class of BLO problems and ensures weak convexity (resp. concavity) of pessimistic (resp. optimistic) hyper-objective functions. Building on this, we show that a zeroth-order algorithm computes approximate Clarke hyper-stationary points with a non-asymptotic convergence guarantee. To the best of our knowledge, this is the first computational guarantee for Clarke-type stationarity for nonsmooth hyper-objective functions in BLO. Our developments, especially the set smoothness property, contribute to a deeper understanding of BLO computability and may inspire applications in other fields.