Efficient Primal Heuristics for Mixed Binary Quadratic Programs Using Suboptimal Rounding Guidance

TL;DR

This paper introduces enhanced primal heuristics for large-scale Mixed Binary Quadratic Programs by leveraging suboptimal relaxation solutions, leading to faster high-quality solutions and reduced primal gaps in complex optimization problems.

Contribution

It extends existing rounding heuristics RENS and Undercover by incorporating suboptimal relaxation solutions to improve solution quality and computational efficiency.

Findings

Significantly reduces primal gap compared to existing heuristics.

Achieves high-quality solutions within small time limits.

Effective on synthetic and real-world MBQP instances.

Abstract

Mixed Binary Quadratic Programs (MBQPs) are a class of NP-hard problems that arise in a wide range of applications, including finance, machine learning, and chemical and energy systems. Large-scale MBQPs are challenging to solve with exact algorithms due to the combinatorial search space and nonlinearity. Primal heuristics have been developed to quickly identify high-quality solutions to challenging combinatorial optimization problems. In this paper, we propose an extension for two well-established rounding-based primal heuristics, RENS and Undercover. Instead of using the optimal solution to a relaxation for variable rounding and search as in RENS, we use a suboptimal relaxation solution of the MBQP as the basis for rounding and guidance for searching over a restricted subproblem where a certain percentage of binary variables are free. We apply a similar idea to the Undercover heuristic that fixes a variable cover to the rounded relaxation values. Instead, we relax a subset of the cover variables based on the suboptimal relaxation and search over a larger restricted subproblem. We evaluate our proposed methods on synthetic MBQP benchmarks and real-world wind farm layout optimization problem instances. The results show that our proposed heuristics identify high-quality solutions within a small time limit and significantly reduce the primal gap and primal integral compared to RENS, Undercover, and solvers with additional primal heuristics integrated inside Branch-and-Bound.