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.
Beyond binarity: Semidefinite programming for ternary quadratic problems
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）；Department of Computer, Control and Management Engineering “Antonio Ruberti”, Sapienza University of Rome（罗马智慧大学，“安东尼奥·鲁贝蒂”计算机、控制与管理工程系）；CentER, Department of Econometrics and OR, Tilburg University（蒂尔堡大学，计量经济学与运筹学系，CentER）；Department of Mathematics “Tullio Levi-Civita”, University of Padua（帕多瓦大学，“图利奥·列维-奇维塔”数学系）
A Tight SDP Relaxation for the Cubic-Quartic Regularization Problem
Xiangtan University（湘潭大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；University of California San Diego（加州大学圣地亚哥分校）；Hong Kong Baptist University（香港浸会大学）
