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arXiv:2409.19668 · INFORMS journal on computing

Local Search for Integer Quadratic Programming

Xiang He, Peng Lin, Tao Jiang, Shaowei Cai

Published May 12, 2026• 0 citations
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Abstract

Integer quadratic programming (IQP) is an important problem in operations research. Local search is a powerful primal heuristic method for solving hard problems, but the research on local search algorithms for solving IQP is still in its early stages. This paper introduces an efficient local search algorithm called LS-IQCQP, which stands for “local search for integer quadratically constrained quadratic programming.” The algorithm is designed for solving general IQP, and it is capable of handling cases where quadratic terms appear in the objective function, constraints, or both. We propose four newly designed operators: the root operator for feasibility and three operators—inequality exploration, equality balance, and unconstrained free operator—for optimizing the objective function. To evaluate the quality of these operations, a weighted scoring function is introduced. Based on the scoring function, LS-IQCQP iteratively applies effective operations and employs a WalkSAT-like approach (a classic local search algorithm for satisfiability problems) by randomly selecting an operation to escape local optima. Experiments are conducted on standard IQP benchmarks from the Quadratic Programming Library (QPLIB) and Mixed-Integer Nonlinear Programming Library (MINLPLIB), comparing LS-IQCQP with several state-of-the-art IQP solvers. Experimental results demonstrate that LS-IQCQP is competitive with the most powerful commercial solver Gurobi and other state-of-the-art solvers in terms of finding a good feasible solution quickly. Moreover, LS-IQCQP has established six new records for QPLIB and MINLPLIB open instances by finding the new best-known solutions. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This research was supported by the National Key Research and Development Program of China [Grant 2023YFA1009500]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1178 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1178 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

DOI
10.1287/ijoc.2025.1178

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Integer quadratic programming (IQP) is an important problem in operations research. Local search is a powerful primal heuristic method for solving hard problems, but the research on local search algorithms for solving IQP is still in its early stages. This paper introduces an efficient local search algorithm called LS-IQCQP, which stands for “local search for integer quadratically constrained quadratic programming.” The algorithm is designed for solving general IQP, and it is capable of handling cases where quadratic terms appear in the objective function, constraints, or both. We propose four newly designed operators: the root operator for feasibility and three operators—inequality exploration, equality balance, and unconstrained free operator—for optimizing the objective function. To evaluate the quality of these operations, a weighted scoring function is introduced. Based on the scoring function, LS-IQCQP iteratively applies effective operations and employs a WalkSAT-like approach (a classic local search algorithm for satisfiability problems) by randomly selecting an operation to escape local optima. Experiments are conducted on standard IQP benchmarks from the Quadratic Programming Library (QPLIB) and Mixed-Integer Nonlinear Programming Library (MINLPLIB), comparing LS-IQCQP with several state-of-the-art IQP solvers. Experimental results demonstrate that LS-IQCQP is competitive with the most powerful commercial solver Gurobi and other state-of-the-art solvers in terms of finding a good feasible solution quickly. Moreover, LS-IQCQP has established six new records for QPLIB and MINLPLIB open instances by finding the new best-known solutions. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This research was supported by the National Key Research and Development Program of China [Grant 2023YFA1009500]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1178 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1178 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .
Quadratic programming (QP) is a well-studied fundamental NP-hard optimization problem which optimizes a quadratic objective over a set of linear constraints. In this paper, we reformulate QPs as a mixed-integer linear problem (MILP). This is done via the reformulation of QP as a linear complementary problem, and the use of binary variables and big-M constraints, to model the complementary constraints. To obtain such reformulation, we show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. Reformulating non-convex QPs as MILPs provides an advantageous way to obtain global solutions as it allows the use of current state-of-the-art MILP solvers. To illustrate this, we compare the performance of our solution approach, labeled quadprogIP, with the current benchmark global QP solver quadprogBB, as well as with BARON, one of the leading non-linear programming (NLP) solvers, and CPLEX's non-convex QP solver, on a large variety of QP test instances. In practice, quadprogIP is shown to typically outperform by orders of magnitude quadprogBB, BARON, and CPLEX on standard QPs. For general QPs, quadprogIP outperforms quadprogBB, outperforms BARON in most instances, while CPLEX performs the best on these instances. For box-constrained QPs, quadprogIP has a comparable performance to quadprogBB and BARON in small- to medium-scale instances, but is outperformed by these solvers on large-scale instances; while CPLEX performs the best on box-constrained QP instances. Also, unlike quadprogBB, the solution approach proposed here is able to solve QP instances whose dual feasible set is unbounded. The MATLAB code, called quadprogIP, and the instances used to perform these numerical experiments are publicly available at https://github.com/xiawei918/quadprogIP.
Binary Quadratic Programs (BQPs) are a class of NP-hard problems that arise in a wide range of applications, including finance, machine learning, and logistics. These problems are challenging to solve due to the combinatorial search space and nonlinearity. In fact, this class of optimization problems is so challenging that, in many instances, standard algorithms struggle to identify feasible solutions within a reasonable time. Primal heuristic algorithms have been developed to quickly identify feasible solutions to BQPs. In this paper, we propose Cover-Relax-Search, an efficient primal heuristic for BQPs. This approach is inspired by multiple local search algorithms, including Undercover. We evaluate the \emph{Cover-Relax-Search} algorithm on multiple BQP benchmarks and show that our proposed heuristic identifies high-quality solutions at a faster speed and significantly reduces the primal integral compared to state-of-the-art solvers and other local search baselines.
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.
This paper presents an iterated Tabu Search(TS) algorithm for the Binary Quadratic Programming(BQP) problem.The TS as the Local Search(LS) and a new perturbation strategy makes the search jump into a more promising area when TS falling into the local optimum.Experimental results show that the proposed algorithm can reach the best known solution on all the 30 large OR-library instances and comparisons with TS.Simulated Annealing(SA) and Memetic Algorithm(MA) demonstrate the competitiveness of the proposed algorithm.
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.
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.