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

Abstract

Quadratic Assignment Problems (QAPs) are known to be among the most challenging discrete optimization problems. Recently, a new class of semi-definite relaxation (SDR) models for QAPs based on matrix splitting has been proposed [25,28]. 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 (SDP). We show that applying the minimal trace principle directly leads to the so-called orthogonal matrix splitting introduced in [28]. 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.

A nonlinear semidefinite optimization relaxation for the worst-case linear optimization under uncertainties

“…In such a case, the SDP solver may fail to find a solution that meets the precision requirement. We also observed similar phenomena in our recent work[29].…”

“…First, to some extent, the robust optimization approach to the WCLO can be interpreted as a tractable convex optimization approximation to the original hard problem. On the other hand, it is well-known that SDO is very successful in providing tight relaxations [23,28,29] and approximate solutions [19,26,32] for many NP-hard optimization problems.…”

A New Semidefinite Programming Relaxation for the Quadratic Assignment Problem and Its Computational Perspectives

“…This limitation has prompted research into exploiting group symmetry of the QAP data matrices A and B to obtain smaller SDP problems; see de Sotirov (2010, 2012). It has also prompted recent research into SDP relaxations of QAP where the matrix variables are of order n; see Mittelmann and Peng (2010) and Peng et al (2010Peng et al ( , 2015. In both these lines of research the authors were able to compute the best-known lower bounds for some QAPLIB instances.…”

“…In particular, we show how one may compute the best-known lower bounds for the unsolved QAPLIB instances Tail35b and Tail40b by combining our new eigenspace relaxation with a known relaxation from Peng et al (2015).…”

Robust quadratic assignment problem with budgeted uncertain flows

“…Many exact and heuristic methods have been developed to solve different cases of QAP. Approximated dynamic programming [10], genetic algorithm [11], parallel algorithms [12,13], hybrid algorithms [14], teaching learning based optimization [15], semidefinite programming relaxations [16,17], QAP with uncertain locations was studied in [39]. [45] used a robust deviation (minmax regret) approach to deal with uncertainty in material flows.…”