On solving integer bilevel optimization problems with a non-convex quadratic follower objective function using disjunctive cuts

2026-10-01

AI Summary

This study addresses integer bilevel optimization problems in which the lower-level objective is a non-convex quadratic function—a class for which existing methods struggle to achieve efficient solutions. To overcome this limitation, we construct an infeasible set based on improving directions and propose a branch-and-cut algorithm. Specifically, disjunctive cuts are generated via linear programming to eliminate infeasible solutions, and a disjunctive term reduction strategy is designed to substantially enhance computational efficiency. Experimental evaluations on both standard convex instances and newly constructed non-convex instances demonstrate that the proposed algorithm significantly outperforms state-of-the-art methods in solution performance. Overall, this work provides an efficient and exact solution framework for such non-convex bilevel optimization problems.

Abstract

In this work, we study bilevel optimization problems where all variables are integer, all constraints and the leader objective function are linear, and the follower objective function is non-convex quadratic. Relying on bilevel-free sets derived from improving directions, we develop a disjunctive cut approach to exclude bilevel-infeasible solutions within a branch-and-cut algorithm. We show that our disjunctive cuts can be obtained by solving a cut generating linear program. Furthermore, we discuss conditions that allow the number of disjuncts in the cut generating linear program to be reduced, and we propose several strategies to identify improving directions and generate disjunctive cuts efficiently. We evaluate various aspects of the proposed branch-and-cut algorithm on both convex instances from the literature that fit our setting and new non-convex instances and compare the performance of our best approach with existing state-of-the-art approaches, which we significantly outperform.

Problem

• integer bilevel optimization
• non-convex quadratic objective
• disjunctive cuts
• branch-and-cut algorithm

Innovation

• Integer bilevel optimization
• Non-convex quadratic objective
• Disjunctive cuts
• Branch-and-cut algorithm
• Bilevel-free sets