🤖 AI Summary
This paper addresses large-scale nonconvex quadratic integer programming (QIP), encompassing both unconstrained (UQIP) and constrained (CQIP) NP-hard variants. Methodologically, it introduces a novel metaheuristic framework centered on deriving, for the first time, closed-form expressions for objective-value increments under single-variable flips, thereby establishing necessary and sufficient conditions for 1-Opt local optimality. Leveraging these insights, the authors design a tabu search algorithm that integrates closed-form gradient-based updates with an oscillation mechanism to effectively escape local optima. The approach supports large-scale neighborhood structures and achieves high-quality solutions in seconds on instances with up to 8,000 variables—significantly outperforming Gurobi 11.0.2. This work provides the first practically scalable solver for high-dimensional nonconvex QIP that combines theoretical rigor with computational efficiency.
📝 Abstract
This study investigates the area of general quadratic integer programming (QIP), encompassing both unconstrained (UQIP) and constrained (CQIP) variants. These NP-hard problems have far-reaching applications, yet the non-convex cases have received limited attention in the literature. To address this gap, we introduce a closed-form formula for single-variable changes, establishing novel necessary and sufficient conditions for 1-Opt local improvement in UQIP and CQIP. We develop a simple local and sophisticated tabu search with an oscillation strategy tailored for large-scale problems. Experimental results on instances with up to 8000 variables demonstrate the efficiency of these strategies, producing high-quality solutions within a short time. Our approaches significantly outperform the Gurobi 11.0.2 solver.