Duality Approach to Bilevel Programs with a Convex Lower Level

TL;DR

This paper introduces a duality-based method for solving bilevel programs with convex lower levels, providing a differentiable dual function, reformulation, and an effective algorithm validated on practical problems.

Contribution

It develops a novel duality approach using partial dualization to create a differentiable dual function and reformulate bilevel programs with convex lower levels.

Findings

The duality-based reformulation is equivalent to the original bilevel problem.

The proposed algorithm effectively solves practical bilevel program instances.

Regularization ensures constraint qualification and algorithm convergence.

Abstract

Bilevel programs are optimization problems where some variables are solutions to optimization problems themselves, and they arise in a variety of control applications, including: control of vehicle traffic networks, inverse reinforcement learning and inverse optimization, and robust control for human-automation systems. This paper develops a duality-based approach to solving bilevel programs where the lower level problem is convex. Our approach is to use partial dualization to construct a new dual function that is differentiable, unlike the Lagrangian dual that is only directionally differentiable. We use our dual to define a duality-based reformulation of bilevel programs, prove equivalence of our reformulation with the original bilevel program, and then introduce regularization to ensure constraint qualification holds. These technical results about our new dual and regularized duality-based reformulation are used to provide theoretical justification for an algorithm we construct for solving bilevel programs with a convex lower level, and we conclude by demonstrating the efficacy of our algorithm by solving two practical instances of bilevel programs.