A Vectorized Positive Semidefinite Penalty Method for Unconstrained Binary Quadratic Programming

The unconstrained binary quadratic programming (UBQP) problem is a class of problems of significant importance in many practical applications, such as in combinatorial optimization, circuit design, and other fields. The positive semidefinite penalty (PSDP) method originated from research on semidefi...

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Published in:Journal of optimization theory and applications Vol. 208; no. 1; p. 53
Main Authors: Huo, Xinyue, Gu, Ran
Format: Journal Article
Language:English
Published: New York Springer US 01.01.2026
Springer Nature B.V
Subjects:
Accuracy
Algorithms
Applications of Mathematics
Branch & bound algorithms
Calculus of Variations and Optimal Control; Optimization
Circuit design
Combinatorial analysis
Design optimization
Efficiency
Engineering
Integer programming
Mathematical programming
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Penalty function
Problem solving
Quadratic programming
Relaxation method (mathematics)
Theory of Computation
Variables
90C20
Positive semidefinite penalty function
65K05
Projection alternating BB stepsize
Binary quadratic programming
Proximal point algorithm
90C09
ISSN:0022-3239, 1573-2878
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Abstract The unconstrained binary quadratic programming (UBQP) problem is a class of problems of significant importance in many practical applications, such as in combinatorial optimization, circuit design, and other fields. The positive semidefinite penalty (PSDP) method originated from research on semidefinite relaxation, where the introduction of an exact penalty function improves the efficiency and accuracy of problem solving. In this paper, we propose a vectorized PSDP method for solving the UBQP problem, which optimizes computational efficiency by vectorizing matrix variables within a PSDP framework. Algorithmic enhancements in penalty updating and initialization are implemented, along with the introduction of two algorithms that integrate the proximal point algorithm and the projection alternating BB method for subproblem resolution. Properties of the penalty function and algorithm convergence are analyzed. Numerical experiments show the superior performance of the method in providing high-quality solutions and satisfactory solution times compared to the semidefinite relaxation method and other established methods.
AbstractList The unconstrained binary quadratic programming (UBQP) problem is a class of problems of significant importance in many practical applications, such as in combinatorial optimization, circuit design, and other fields. The positive semidefinite penalty (PSDP) method originated from research on semidefinite relaxation, where the introduction of an exact penalty function improves the efficiency and accuracy of problem solving. In this paper, we propose a vectorized PSDP method for solving the UBQP problem, which optimizes computational efficiency by vectorizing matrix variables within a PSDP framework. Algorithmic enhancements in penalty updating and initialization are implemented, along with the introduction of two algorithms that integrate the proximal point algorithm and the projection alternating BB method for subproblem resolution. Properties of the penalty function and algorithm convergence are analyzed. Numerical experiments show the superior performance of the method in providing high-quality solutions and satisfactory solution times compared to the semidefinite relaxation method and other established methods.
ArticleNumber 53
Author Gu, Ran
Huo, Xinyue
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Keywords 90C20
Positive semidefinite penalty function
65K05
Projection alternating BB stepsize
Binary quadratic programming
Proximal point algorithm
90C09
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Snippet The unconstrained binary quadratic programming (UBQP) problem is a class of problems of significant importance in many practical applications, such as in...
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SubjectTerms Accuracy
Algorithms
Applications of Mathematics
Branch & bound algorithms
Calculus of Variations and Optimal Control; Optimization
Circuit design
Combinatorial analysis
Design optimization
Efficiency
Engineering
Integer programming
Mathematical programming
Mathematics
Mathematics and Statistics
Methods
Operations Research/Decision Theory
Optimization
Penalty function
Problem solving
Quadratic programming
Relaxation method (mathematics)
Theory of Computation
Variables
Title A Vectorized Positive Semidefinite Penalty Method for Unconstrained Binary Quadratic Programming
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