ADMM for the SDP relaxation of the QAP

4 The semidefinite programming SDP relaxation has proven to be extremely strong for many hard 5 discrete optimization problems. This is in particular true for the quadratic assignment problem QAP, 6 arguably one of the hardest NP-hard discrete optimization problems. There are several difficulties that 7 arise in efficiently solving the SDP relaxation, e.g., increased dimension; inefficiency of the current primal8 dual interior point solvers in terms of both time and accuracy; and difficulty and high expense in adding 9 cutting plane constraints. 10 We propose using the alternating direction method of multipliers ADMM to solve the SDP relaxation. 11 This first order approach allows for inexpensive iterations, a method of cheaply obtaining low rank solu12 tions, as well a trivial way of adding cutting plane inequalities. When compared to current approaches 13 and current best available bounds we obtain remarkable robustness, efficiency and improved bounds. 14