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

Gioni Mexi, Deborah Hendrych, Sébastien Designolle, Mathieu Besançon, Sebastian Pokutta

TL;DR

The paper addresses nonconvex MIQCQPs by extending the Boscia Frank-Wolfe-based primal solver into a practical heuristic framework. It combines problem transformations, a power-penalty relaxation, and a rich set of primal heuristics—including rounding, gradient-driven search, and large neighborhood search leveraging FW-generated integer vertices—to rapidly find high-quality feasible solutions and occasionally derive dual insights. Key contributions include reformulating nonlinearities into the objective, handling complementarities and perspective constraints, and introducing ASENS, Undercover, and RINS variants alongside targeted QUBO heuristics, all aided by parallel restarts. Empirical results on QPLIB show the method achieving first place in the Land-Doig MIP Competition 2025 and delivering eight new best-known solutions within five minutes, highlighting its practical impact for challenging MIQCQPs.

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.

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

TL;DR

The paper addresses nonconvex MIQCQPs by extending the Boscia Frank-Wolfe-based primal solver into a practical heuristic framework. It combines problem transformations, a power-penalty relaxation, and a rich set of primal heuristics—including rounding, gradient-driven search, and large neighborhood search leveraging FW-generated integer vertices—to rapidly find high-quality feasible solutions and occasionally derive dual insights. Key contributions include reformulating nonlinearities into the objective, handling complementarities and perspective constraints, and introducing ASENS, Undercover, and RINS variants alongside targeted QUBO heuristics, all aided by parallel restarts. Empirical results on QPLIB show the method achieving first place in the Land-Doig MIP Competition 2025 and delivering eight new best-known solutions within five minutes, highlighting its practical impact for challenging MIQCQPs.

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.

Paper Structure

This paper contains 29 sections, 12 equations, 4 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Boscia: Integer Frank-Wolfe
  3. Problem transformations and presolving
  4. Propagation
  5. Power penalty
  6. Quadratic special structures
  7. Complementarities.
  8. Perspective constraints.
  9. Convexification of quadratic binary problems
  10. Primal Heuristics
  11. Rounding
  12. Gradient-based heuristics
  13. Large neighborhood search
  14. Active Set Enforced Neighborhood Search.
  15. Undercover.
  16. ...and 14 more sections

Figures (4)

  • Figure 1: Overview of our approach. The model first undergoes a pre-processing whose main goal is to transfer nonlinearities into the objective function. Our workhorse is indeed the Frank-Wolfe-based solver Boscia which handles nonlinear problems (NLP) by suitably combining calls to a mixed-integer programming (MIP) solver in a branch-and-bound (BnB) framework. The points collected at each node in this process are then fed to various heuristics to reach better solutions.
  • Figure 2: Distribution of the optimality gap, and primal integral over MIQCQPs where a solution was found by at least one of the settings.
  • Figure 3: Distribution of the optimality gap, and primal integral over MIQCQPs where a solution was found by at least one of the settings.
  • Figure 4: Impact of the RINS heuristic on the performance of the base heuristic across MIQCQPs where a solution was found by at least one of the settings.