Set Smoothness Unlocks Clarke Hyper-stationarity in Bilevel Optimization

He Chen, Jiajin Li, Anthony Man-Cho So

Abstract:

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 struc
tural 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 guar
antee. 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.