This paper deals with several combinatorial optimization problems. The most challenging such problem is the quadratic assignment problem. It is considered in both two dimensions (QAP) and in three dimensions (Q3AP) and in the context of communication engineering. Semidefinite relaxations are used to derive lower bounds for the optimum while heuristics are applied to either find upper bounds or good feasible solutions. Semidefinite relaxations also yield bounds for questions related to binary and spherical codes including for the kissing number problem. Finally, two combinatorial problems are solved exactly, a Q3AP from communications and a directional sensor location problem.