On Stability in Optimistic Bilevel Optimization

TL;DR

This work tackles instability in optimistic bilevel optimization by introducing a lifted formulation $(P)^ u$ that remains well-posed under mild, nonconvex and nonsmooth conditions. It establishes bounds such as $\limsup_{\nu\to\infty} \mathfrak{m}^\nu \le \mathfrak{m}$ and, under calmness at a point or locally, proves convergence of near-optimal solutions of $(P)^\nu$ to optimal solutions of $(P)$, often via exact penalization. An outer-approximation algorithm is developed for solving $(P)^\nu$, and illustrative examples underscore the stability gains when lower-level problems include integers or disjunctions. The results offer a robust framework for attainability and computation in complex bilevel settings, without requiring convexity or smoothness.

Abstract

Solutions of bilevel optimization problems tend to suffer from instability under changes to problem data. In the optimistic setting, we construct a lifted formulation that exhibits desirable stability properties under mild assumptions that neither invoke convexity nor smoothness. The upper- and lower-level problems might involve integer restrictions and disjunctive constraints. In a range of results, we invoke at most pointwise and local calmness for the lower-level problem in a sense that holds broadly. The lifted formulation is computationally attractive with structural properties being brought out and an outer approximation algorithm becoming available.

Theorems & Definitions (19)

• Theorem 2.2
• Example 2.3
• Definition 2.4
• Proposition 2.5
• Theorem 2.6
• Example 2.7
• Proposition 2.8
• Definition 2.9
• Theorem 2.10
• Example 3.1