Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

RA Journal Of Applied Research

Improved SDP Bounds for Binary Quadratic Programming

Baiyi Wu

Published Jun 27, 2018• 0 citations
Publisher

Abstract

As a classical combinatorial problem, the binary quadratic programming problem has many applications in finance, statistics, production management, etc. The state-of-the-art solution for solving this problem accurately is based on branch-and-bound frameworks, with the low bound support of a semi-definite programming (SDP) relaxation. This paper generalizes the spectral bounds in the literature and proposes a sequence of improved SDP bounds for the binary quadratic programming problem. Our method relies on the closest binary points to an affine space, which can be found by reverse enumeration technique.

DOI
10.31142/rajar/v4i6.09

Related papers

A fast semidefinite approach to solving binary quadratic problems

Peng Wang, Chunhua Shen, Anton van den Hengel

Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spec-tral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high for large scale prob-lems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, com-pared with conventional SDP methods, the new SDP for-mulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmen-tation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs. 1.

•Jan 1, 2013•26 citations

Large-Scale Binary Quadratic Optimization Using Semidefinite Relaxation and Applications

Peng Wang, Chunhua Shen, Anton van den Hengel +1 more

In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.

arXiv:1411.7564•IEEE Transactions on Pattern Analysis and Machine Intelligence•Mar 11, 2016•59 citations

A Direct SDP Approach for Constrained Quadratic Binary Optimization Problems

MilaNedic, Zvmcevap

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.

•Aug 25, 2026•0 citations

QUBO Dual Bounds via SDP Plane Projection Method

Apostolos Chalkis, Thomas Kleinert, Boro Sofranac

In this paper, we present a new method to solve a certain type of Semidefinite Programming (SDP) problems. These types of SDPs naturally arise in the Quadratic Convex Reformulation (QCR) method and can be used to obtain dual bounds of Quadratic Unconstrained Binary Optimization (QUBO) problems. QUBO problems have recently become the focus of attention in the quantum computing and optimization communities as they are well suited to both gate-based and annealing-based quantum computers on one side, and can encompass an exceptional variety of combinatorial optimization problems on the other. Our new method can be easily warm-started, making it very effective when embedded into a branch-and-bound scheme and used to solve the QUBO problem to global optimality. We test our method in this setting and present computational results showing the effectiveness of our approach.

arXiv:2312.15026•Dec 22, 2023•1 citation

Quadratic Combinatorial Optimization Using Separable Underestimators

Christoph Buchheim, Emiliano Traversi

Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binary constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems to obtain a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both CPLEX and the related QCR method as well as the SDP-based software BiqCrunch on instances of the quadratic shortest path problem and the quadratic assignment problem.

INFORMS journal on computing•Aug 1, 2018•17 citations

A SUCCESSIVE QUADRATIC PROGRAMMING ALGORITHM FOR SDP RELAXATION OF THE BINARY QUADRATIC PROGRAMMING

Xuewen Mu, SANYANG LID, Yaling Zhang

In this paper, we obtain a successive quadratic programming algorithm for solving the semidefinite programming (SDP) relaxation of the binary quadratic programming. Combining with a randomized method of Goemans and Williamson, it provides an efficient approximation for the binary quadratic programming. Furthermore, its convergence result is given. At last, We report some numerical examples to compare our method with the interior-point method on Maxcut problem.

Bulletin of the Korean Mathematical Society•Nov 1, 2005•0 citations
As a classical combinatorial problem, the binary quadratic programming problem has many applications in finance, statistics, production management, etc. The state-of-the-art solution for solving this problem accurately is based on branch-and-bound frameworks, with the low bound support of a semi-definite programming (SDP) relaxation. This paper generalizes the spectral bounds in the literature and proposes a sequence of improved SDP bounds for the binary quadratic programming problem. Our method relies on the closest binary points to an affine space, which can be found by reverse enumeration technique.
Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spec-tral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high for large scale prob-lems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, com-pared with conventional SDP methods, the new SDP for-mulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmen-tation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs. 1.
In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.
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.
In this paper, we present a new method to solve a certain type of Semidefinite Programming (SDP) problems. These types of SDPs naturally arise in the Quadratic Convex Reformulation (QCR) method and can be used to obtain dual bounds of Quadratic Unconstrained Binary Optimization (QUBO) problems. QUBO problems have recently become the focus of attention in the quantum computing and optimization communities as they are well suited to both gate-based and annealing-based quantum computers on one side, and can encompass an exceptional variety of combinatorial optimization problems on the other. Our new method can be easily warm-started, making it very effective when embedded into a branch-and-bound scheme and used to solve the QUBO problem to global optimality. We test our method in this setting and present computational results showing the effectiveness of our approach.
Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binary constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems to obtain a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both CPLEX and the related QCR method as well as the SDP-based software BiqCrunch on instances of the quadratic shortest path problem and the quadratic assignment problem.