Enhancing Semidefinite Relaxation for Quadratically Constrained Quadratic Programming via Penalty Methods

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...

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Vydáno v:Journal of optimization theory and applications Ročník 180; číslo 3; s. 964 - 992
Hlavní autoři: Luo, Hezhi, Bai, Xiaodi, Peng, Jiming
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.03.2019
Springer Nature B.V
Témata:
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
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Popis
Shrnutí: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.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1416-0