An Alternating Primal Heuristic for Nonconvex MIQCQP with Dynamic Convexification and Parallel Local Branching

TL;DR

This paper introduces a new primal heuristic for nonconvex MIQCQPs that uses dynamic convexification and parallel local branching, achieving improved solutions on benchmark instances.

Contribution

It presents a novel alternating heuristic with dynamic convex approximation and parallel local branching, advancing solution quality for challenging MIQCQPs.

Findings

Solved three previously unsolved instances from QPLIB.

Improved best-known solutions for fifteen instances.

Validated effectiveness within five minutes runtime.

Abstract

We develop a novel primal heuristic for nonconvex Mixed-Integer Quadratically Constrained Quadratic Programs (MIQCQPs). The method is built around a convex approximation that is dynamically adjusted within a feasibility-pump-style alternating heuristic. Approximations are adjusted based on the structure of the MIQCQP instance. Additionally, parallelized local branching is incorporated to further refine detected solutions. This paper builds upon the second-place finalist submission in the 2025 Land-Doig MIP Computational Competition. Our results are validated with computational experiments on instances from QPLIB, finding feasible solutions for three previously unsolved cases and improving the best-known solutions for fifteen instances within five minutes of runtime.