Proximity-based approximation algorithms for integer bilevel programs

TL;DR

This paper investigates the complexity of bilevel programs with convex quadratic lower levels under integer constraints, establishing hardness results and proposing proximity-based algorithms that offer additive approximation guarantees, supported by computational evidence.

Contribution

It introduces proximity-based approximation algorithms for integer bilevel programs and proves their additive approximation bounds, extending results to cases with integer linear followers.

Findings

The problem is $\Sigma_2^p$-hard to decide.

The proposed algorithm provides additive approximation guarantees.

Computational experiments show practical speed and solution quality benefits.

Abstract

We primarily consider bilevel programs where the lower level is a convex quadratic minimization problem under integer constraints. We show that it is $\Sigma_2^p$-hard to decide if the optimal objective for the leader is lesser than a given value. Following that, we consider a natural algorithm for bilevel programs that is used as a heuristic in practice. Using a result on proximity in convex quadratic minimization, we show that this algorithm provides an additive approximation to the optimal objective value of the leader. The additive constant of approximation depends on the flatness constant corresponding to the dimensionality of the follower's decision space and the condition number of the matrix $Q$ defining the quadratic term in the follower's objective function. We show computational evidence indicating the speed advantage as well as that the solution quality guarantee is much better than the worst-case bounds. We extend these results to the case where the follower solves an integer linear program, with an objective perfectly misaligned with that of the leader. Using integer programming proximity, we show that a similar algorithm, used as a heuristic in the literature, provides additive approximation guarantees.