Enhancing RLT-based relaxations for polynomial programming problems via a new class of v-semidefinite cuts

• Published in: Computational optimization and applications Vol. 52; no. 2; pp. 483 - 506
• Main Authors: Sherali, Hanif D., Dalkiran, Evrim, Desai, Jitamitra
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01-06-2012; Springer Nature B.V
• Subjects: Algorithms; Analysis; Convex and Discrete Geometry; Cutting; Eigenvalues; Linear programming; Management Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices; Matrix methods; Operations Research; Operations Research/Decision Theory; Optimization; Planes; Polynomials; Programming; Semidefinite programming; Statistics; Strategy; Studies; Systems engineering; Tightening; Polynomial programs; Semidefinite programming; BARON; Global optimization; Semidefinite cuts; Reformulation-Linearization Technique (RLT); GloptiPoly
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-011-9425-z

In this paper, we propose to enhance Reformulation-Linearization Technique (RLT)-based linear programming (LP) relaxations for polynomial programming problems by developing cutting plane strategies using concepts derived from semidefinite programming. Given an RLT relaxation, we impose positive semidefiniteness on suitable dyadic variable-product matrices, and correspondingly derive implied semidefinite cuts. In the case of polynomial programs, there are several possible variants for selecting such particular variable-product matrices on which positive semidefiniteness restrictions can be imposed in order to derive implied valid inequalities. This leads to a new class of cutting planes that we call
v
-semidefinite cuts. We explore various strategies for generating such cuts, and exhibit their relative effectiveness towards tightening the RLT relaxations and solving the underlying polynomial programming problems in conjunction with an RLT-based branch-and-cut scheme, using a test-bed of problems from the literature as well as randomly generated instances. Our results demonstrate that these cutting planes achieve a significant tightening of the lower bound in contrast with using RLT as a stand-alone approach, thereby enabling a more robust algorithm with an appreciable reduction in the overall computational effort, even in comparison with the commercial software BARON and the polynomial programming problem solver GloptiPoly.