Semi-definite programming relaxation of quadratic assignment problems based on nonredundant matrix splitting

Quadratic assignment problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation models for QAPs based on matrix splitting has been proposed (Mittelmann and Peng, SIAM J Optim 20:3408–3426, <CitationRef CitationID="CR25">2010</CitationRef>; Peng et al., Math Program Comput 2:59–77, <CitationRef CitationID="CR27">2010</CitationRef>). In this paper, we consider the issue of how to choose an appropriate matrix splitting scheme so that the resulting relaxation model is easy to solve and able to provide a strong bound. For this, we first introduce a new notion of the so-called redundant and non-redundant matrix splitting and show that the relaxation based on a non-redundant matrix splitting can provide a stronger bound than a redundant one. Then we propose to follow the minimal trace principle to find a non-redundant matrix splitting via solving an auxiliary semi-definite programming problem. We show that applying the minimal trace principle directly leads to the so-called orthogonal matrix splitting introduced in (Peng et al., Math Program Comput 2:59–77, <CitationRef CitationID="CR27">2010</CitationRef>). To find other non-redundant matrix splitting schemes whose resulting relaxation models are relatively easy to solve, we elaborate on two splitting schemes based on the so-called one-matrix and the sum-matrix. We analyze the solutions from the auxiliary problems for these two cases and characterize when they can provide a non-redundant matrix splitting. The lower bounds from these two splitting schemes are compared theoretically. Promising numerical results on some large QAP instances are reported, which further validate our theoretical conclusions. Copyright Springer Science+Business Media New York 2015