Topics in matrix analysis
A new lower bound via projection for the quadratic assignment problem
Mathematics of Operations Research
Lower bounds for the quadratic assignment problem via triangle decompositions
Mathematical Programming: Series A and B
QAPLIB – A Quadratic Assignment ProblemLibrary
Journal of Global Optimization
Lower Bounds for the Quadratic Assignment Problem Based Upon a Dual Formulation
Operations Research
Solving Lift-and-Project Relaxations of Binary Integer Programs
SIAM Journal on Optimization
Bounds for the quadratic assignment problem using the bundle method
Mathematical Programming: Series A and B
Mathematical Programming: Series A and B
A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem
Mathematics of Operations Research
Bounds on the performance of vector-quantizers under channel errors
IEEE Transactions on Information Theory
Copositive and semidefinite relaxations of the quadratic assignment problem
Discrete Optimization
On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming
Mathematics of Operations Research
A distance sum-based hybrid method for intrusion detection
Applied Intelligence
Hi-index | 0.00 |
Quadratic assignment problems (QAPs) with a Hamming distance matrix for a hypercube or a Manhattan distance matrix for a rectangular grid arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes of QAPs based on semidefinite programming (SDP). By exploiting the data structure of the distance matrix $B$, we first show that for any permutation matrix $X$, the matrix $Y=\alpha E-XBX^T$ is positive semidefinite, where $\alpha$ is a properly chosen parameter depending only on the associated graph in the underlying QAP and $E=ee^T$ is a rank-1 matrix whose elements have value 1. This results in a natural way to approximate the original QAPs via SDP relaxation based on the matrix-splitting technique. Our new SDP relaxations have a smaller size compared with other SDP relaxations in the literature and can be solved efficiently by most open source SDP solvers. Experimental results show that for the underlying QAPs of size up to $n=200$, strong bounds can be obtained effectively.