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

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 functions 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.