Minimum energy configurations on a toric lattice as a quadratic assignment problem

We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex programming (CQP), and semidefinite (SDP) bound. Since last two were not compared directly before, we prove that SDP bound is stronger than CQP then apply these to improve on discrete energy minimization problem, reformulated as QAP, which aims minimize potential between repulsive particles toric grid. Thus are able optimality several configurations of grid sizes, complementing earlier results by Bouman, Draisma Van Leeuwaarden [ SIAM Journal Discrete Mathematics, 27(3):1295--1312, 2013]. The programs in question too large solve without pre-processing, use symmetry reduction method Parrilo Permenter [Mathematical Programming, 181:51--84, 2020] make computation possible.