In this paper, we show that the quadratic assignment problem (QAP) can be reformulated to an equivalent rank constrained doubly nonnegative (DNN) problem. Under the framework of the difference of convex functions (DC) approach, a semi-proximal DC algorithm is proposed for solving the relaxation of the rank constrained DNN problem whose subproblems can be solved by the semi-proximal augmented Lagrangian method. We show that the generated sequence converges to a stationary point of the corresponding DC problem, which is feasible to the rank constrained DNN problem under some suitable assumptions. Moreover, numerical experiments demonstrate that for most QAP instances, the proposed approach can find the global optimal solutions efficiently, and for others, the proposed algorithm is able to provide good feasible solutions in a reasonable time.

在本文中，我们表明可以将二次分配问题（QAP）重构为等效秩约束双非负（DNN）问题。在凸函数差分法的框架下，提出了一种半近似DC算法来解决秩受限DNN问题的松弛问题，该子问题可以通过半近似增强拉格朗日方法求解。我们表明，生成的序列收敛到相应DC问题的平稳点，这在某些合适的假设下对于秩受限DNN问题是可行的。此外，数值实验表明，对于大多数QAP实例，所提出的方法可以有效地找到全局最优解，而对于其他QAP实例，所提出的算法则能够在合理的时间内提供良好的可行解。