Obtaining an Approximate Solution for Quadratic Maximization Problems
In this paper, we consider indefinite quadratic maximization problems over inequality constraints. Through the Reformulation and Linearization Technique (RLT), we reformulate the problem as a linear maximization problem over a region which is given by the convex hull of a nonconvex region. A crucial point of this reformulation is how to describe a tight relaxation of the convex hull efficiently. While, in the standard RLT procedure, we generate valid linear inequalities for the convex hull by taking the pairwise products of the original inequality constraints, we propose a new class of linear inequalities that are derived from the polyhedral structure of the cut polytope. In general, these inequalities are not implied by those generated by the pairwise products, and thus generate a tighter relaxation of the convex hull.
We also give results on our numerical experiments over general quadratic maximization problems. We show that a cutting plane procedure which is incorporated with the positive semi-definite constraints generates an almost exact optimal solution of nonconvex problems with up to 110 variables.
Keywordsnonconvex quadratic programs semi-definite programming cutting plane method
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