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Journal of Global Optimization

An improved lower bound and approximation algorithm for binary constrained quadratic programming problem

Cheng Zhu Lu, Zhenbo Wang, Wenxun Xing

Published Dec 17, 2009• 6 citations
Publisher
DOI
10.1007/s10898-009-9504-1

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•Aug 25, 2026•0 citations
The multi-linearly constrained separable binary quadratic programming problem (MLSBQP) is a direct extension of the classical singly-linearly constrained separable binary quadratic programming problem (SLSBQP). In the literature, several algorithms have been proposed for SLSBQP which can solve problem instances with millions of variables globally. However, for the case of MLSBQP, the current state-of-the-art solver can only solve problems with up to 40000 binary variables within a reasonable computational time. In this paper, we propose an extremely efficient branch-and-bound algorithm for solving MLSBQP. The main feature of the proposed algorithm is that an efficient dual coordinate ascent algorithm is applied to obtain lower bounds, and the complexity of each inner loop of the dual coordinate ascent algorithm is $ \mathcal{O}(n\log n) $. Our numerical experiments show that the proposed branch-and-bound algorithm can solve MLSBQP with up to millions of variables within only several minutes, and performs much faster than Gurobi when the number of linear constraints is small.
This repository contains the source code, computational experiments, analysis, and benchmarks accompanying the paper A Direct SDP-Based Framework for Certified Bounds for Constrained Quadratic Binary Optimization Problems. The repository implements a direct semidefinite programming (SDP) approach for constrained quadratic binary optimization, designed to obtain strong bounds for exact optimization. The SDP relaxation is strengthened using quadratized linear constraints and triangle (hypermetric) inequalities. To address the numerical infeasibility of SDP solutions obtained from numerical solvers such as MOSEK, the framework includes a correction procedure that converts approximate SDP solutions into mathematically certified bounds suitable for use in Branch-and-Bound algorithms. The repository also implements an adaptation of the Goemans–Williamson randomized hyperplane rounding scheme to constrained binary optimization, together with problem-specific feasibility restoration. Computational experiments are provided for the Sparsest k-Subgraph and Maximum Independent Set problems. The work presented in this repository was funded and developed under the QBIQ project (project no. 644), funded by the Slovenian Research and Innovation Agency (ARIS) through the Applied Projects programme. The code and accompanying data are intended to facilitate reproducibility of the computational results and provide a foundation for further research on exact SDP-based methods for constrained quadratic binary optimization.