A Matrix-lifting Semidefinite Relaxation for the Quadratic Assignment Problem

The quadratic assignment problem (\QAP) is arguably one of the hardest of the NP-hard discrete optimization problems. Problems of dimension greater than 20 are considered to be large scale. Current successful solution techniques depend on branch and bound methods. However, it is difficult to get \emph{strong and inexpensive} bounds. In this paper we introduce a … Read more

Copositive and Semidefinite Relaxations of the Quadratic Assignment Problem

Semidefinite relaxations of the quadratic assignment problem (QAP) have recently turned out to provide good approximations to the optimal value of QAP. We take a systematic look at various conic relaxations of QAP. We first show that QAP can equivalently be formulated as a linear program over the cone of completely positive matrices. Since it … Read more

Bounds for the Quadratic Assignment Problem Using the Bundle Method

Semidefinite Programming (SDP) has recently turned out to be a very powerful tool for approximating some NP-hard problems. The nature of the Quadratic Assignment Problem suggests SDP as a way to derive tractable relaxation. We recall some SDP relaxations of QAP and solve them approximately using the Bundle Method. The computational results demonstrate the efficiency … Read more

The Steinberg Wiring Problem

In 1961 Leon Steinberg formulated a “backboard wiring” problem that has resisted solution for 40 years. Steinberg’s wiring problem is to determine the locations of 34 computer components on a 4 by 9 grid so as to minimize the total length of the wiring required to interconnect them. The problem is an example of a … Read more

Optimization on Computational Grids

We define the concept of a computational grid, and describe recent work in solving large and complex optimization problems on this type of platform; in particular, integer programming, the quadratic assignment problem, and stochastic programming problems. This article focuses on work conducted in the metaneos project. Citation Preprint, Mathematics and Computer Science Division, Argonne National … Read more