On Stability in Optimistic Bilevel Optimization

TL;DR

This paper introduces a lifted formulation for optimistic bilevel optimization that enhances stability under data perturbations, even with complex constraints, and provides an efficient outer approximation algorithm.

Contribution

It presents a novel lifted formulation for bilevel problems that ensures stability without requiring convexity or smoothness, applicable to problems with integer and disjunctive constraints.

Findings

Lifted formulation exhibits desirable stability properties.

Outer approximation algorithm is computationally attractive.

Applicable to problems with integer restrictions and disjunctive constraints.

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.