A parametric integer programming algorithm for bilevel mixed integer programs

TL;DR

This paper introduces polynomial time algorithms for solving certain bilevel mixed integer programs using parametric integer programming, addressing cases with continuous leader variables and fixed total variables.

Contribution

It develops new polynomial time algorithms for bilevel mixed integer problems, including detection of unattainable infimum costs and approximation schemes.

Findings

Algorithms run in polynomial time for fixed total variables.

Detects when infimum cost is not attained.

Provides approximation schemes with logarithmic precision complexity.

Abstract

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.