# Semidefinite Programming Relaxations For The Quadratic Assignment Problem

## Background

The quadratic assignment problem (QAP) is one of the hardest NP-hard discrete optimization problems. The semidefinite programming (SDP) relaxation has proven to be extremely strong for QAP by lifting the variable [1]. However, the SDP relaxation for the QAP becomes very large, and traditional methods such as the interior-point method can hardly solve problems of even medium size, say 30.

We reformulate and also strengthen the relaxation model in [1] by splitting variables and adding more constraints. Then we employ the alternating direction method of multiplier (ADMM) to solve the strengthened model with or without an additional low-rank constraint. We obtain very sharp lower bound on benchmark datasets within much shorter time compared to the best existing method.

## Notation and our method

The QAP problem can be formulated as

where and are real symmetric matrices, is a real matrix, and denotes the set of permutation matrices. By lifting variable, [1] relaxed the original QAP problem into

where is a sampling operator (also called gangster operator) picking the elements in the index set , is a matrix with all zeros except the top-left one, is a basis matrix (see our report), and

We introduce another variable and enforce and also restrict , resulting in the following model

Then the ADMM method is applied to solve the above model with or without the rank-one constraint . Due to the introducing of , every step in the ADMM method has closed form solution, and numerically ADMM is very efficient on solving our problem.

## Numerical results

We test our method on 45 benchmark QAP instances. Compared to the best existing lower bound (column 2 by Bundle method [2]), our method improves the lower bound on every instance. In addition, our method is very fast, in particular when there is a low-rank constraint (column 9).

## Matlab code

Download the code

## Citation

*D. Oliveira, H. Wolkowicz, and Y. Xu*. ADMM for the SDP relaxation of the QAP. arXiv1512.05448, 2015.

## References

. Q. Zhao, S.E. Karisch, F. Rendl, and H. Wolkowicz. *Semidefinite programming relaxations for the quadratic assignment problem*, J. Combinatorial Optimization, 2(1): 71–109, 1998.

. F. Rendl and R. Sotirov. Bounds for the quadratic assignment problem using the bundle method. Math. Program., 109 (2–3, Ser. B): 505–524, 2007

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A new semidefinite programming, SDP, relaxation for the *general* graph partitioning problem, GP, is derived. The relaxation arises from the dual of the (homogenized) Lagrangian dual of an appropriate quadratic representation of GP. The quadratic representation includes a representation of the 0,1 constraints in GP. The special structure of the relaxation is exploited in order to project onto the *minimal face* of the cone of positive-semidefinite matrices which contains the feasible set. This guarantees that the Slater constraint qualification holds, which allows for a numerically stable primalâ€“dual interior-point solution technique. A *gangster operator* is the key to providing an efficient representation of the constraints in the relaxation. An incomplete preconditioned conjugate gradient method is used for solving the large linear systems which arise when finding the Newton direction. Only dual feasibility is enforced, which results in the desired lower bounds, but avoids the expensive primal feasibility calculations. Numerical results illustrate the efficacy of the SDP relaxations.