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Mathematical Programming Computation

QPLIB: a library of quadratic programming instances

Fabio Furini, Emiliano Traversi, Pietro Belotti, Antonio Frangioni, Ambros Gleixner, Nick Gould, Leo Liberti, Andrea Lodi, Ruth Misener, Hans Detlef Mittelmann, Nikolaos V. Sahinidis, Stefan Vigerske, Angelika Wiegele

Published Sep 22, 2018• 89 citations
Publisher
DOI
10.1007/s12532-018-0147-4

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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/ .
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.
Machine learning-based optimization frameworks have attracted increasing attention for accelerating the solution of large-scale quadratically constrained quadratic programs (QCQPs) by exploiting shared problem structure across instances. However, existing machine learning (ML) frameworks often rely on the assumption of parametric models and large-scale solvers. This article introduces HEQP, a hypergraph neural network-based evolutionary optimization framework for large-scale QCQPs. This framework features two main components: 1) hypergraph-based neural prediction, which predicts optimal solutions for QCQPs without assumptions of models; and 2) evolutionary large neighborhood search (Evo-LNS), which employs a McCormick relaxation-based repair strategy to search and apply crossover on neighborhood solutions using a small-scale solver. We further show that our framework is equivalent to the interior-point method (IPM), a polynomial-time algorithm, for quadratic programming. Experiments on two types of benchmark problems and 13 large-scale real-world instances from the QPLIB illustrate that our framework outperforms state-of-the-art solvers (including Gurobi, SCIP, and SHOT) in both solution quality and time efficiency, highlighting the efficiency of ML-based optimization frameworks for QCQPs.