We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman et al. (2013). The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Permenter and Parrilo (2020) to make computation of the SDP bounds possible.

我们考虑二次分配问题（QAP）的三个已知边界：特征值，凸二次规划（CQP）和半定规划（SDP）边界。由于最后两个边界之前没有直接进行比较，因此我们证明SDP边界比CQP边界更强。然后，我们将其应用于改进离散能量最小化问题的已知范围，该问题被重新格式化为QAP，旨在最小化复曲面网格上排斥粒子之间的势能。因此，我们有能力证明几种颗粒和网格尺寸配置的最优性，补充了Bouman等人的早期结果。（2013）。所讨论的半定程序太大，无法进行预处理，因此我们使用Permenter和Parrilo（2020）的对称性减少方法来计算SDP边界。