Enhancing Semidefinite Relaxation for Quadratically Constrained Quadratic Programming via Penalty Methods

• Published in: Journal of optimization theory and applications Vol. 180; no. 3; pp. 964 - 992
• Main Authors: Luo, Hezhi, Bai, Xiaodi, Peng, Jiming
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2019; Springer Nature B.V
• Subjects: Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Engineering; Iterative methods; Linear programming; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Penalty function; Quadratic programming; Theory of Computation; Conditionally quasi-convex relaxation; 90C20; Bisection search; 90C22; Iterative search; 90C26; Quadratic programming; Penalty method
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-018-1416-0

Quadratically constrained quadratic programming arises from a broad range of applications and is known to be among the hardest optimization problems. In recent years, semidefinite relaxation has become a popular approach for quadratically constrained quadratic programming, and many results have been reported in the literature. In this paper, we first discuss how to assess the gap between quadratically constrained quadratic programming and its semidefinite relaxation. Based on the estimated gap, we discuss how to construct an exact penalty function for quadratically constrained quadratic programming based on its semidefinite relaxation. We then introduce a special penalty method for quadratically constrained linear programming based on its semidefinite relaxation, resulting in the so-called conditionally quasi-convex relaxation. We show that the conditionally quasi-convex relaxation can provide tighter bounds than the standard semidefinite relaxation. By exploring various properties of the conditionally quasi-convex relaxation model, we develop two effective procedures, an iterative procedure and a bisection procedure, to solve the constructed conditionally quasi-convex relaxation. Promising numerical results are reported.