A Frank-Wolfe-based primal heuristic for quadratic mixed-integer optimization

TL;DR

This paper introduces a novel primal heuristic based on the Frank-Wolfe algorithm for solving challenging quadratic mixed-integer problems, extending existing convex methods to nonconvex cases with significant computational success.

Contribution

It extends the Boscia framework to nonconvex quadratic problems, incorporating reformulations, heuristics, and neighborhood searches to improve solution quality and efficiency.

Findings

Achieved significant improvements on QPLIB instances

Solved challenging MIQCQPs within minutes

Won first place in the Land-Doig MIP Competition 2025

Abstract

We propose a primal heuristic for quadratic mixed-integer problems. Our method extends the Boscia framework -- originally a mixed-integer convex solver leveraging a Frank-Wolfe-based branch-and-bound approach -- to address nonconvex quadratic objective and constraints. We reformulate nonlinear constraints, introduce preprocessing steps, and a suite of heuristics including rounding strategies, gradient-guided selection, and large neighborhood search techniques that exploit integer-feasible vertices generated during the Frank-Wolfe iterations. Computational results demonstrate the effectiveness of our method in solving challenging MIQCQPs, achieving improvements on QPLIB instances within minutes and winning first place in the Land-Doig MIP Computational Competition 2025.