Semidefinite approximations for quadratic programs over orthogonal matrices

Finding global optimum of a non-convex quadratic function is in general a very difficult task even when the feasible set is a polyhedron. We show that when the feasible set of a quadratic problem consists of orthogonal matrices from Rn×k, then we can transform it into a semidefinite program in matrices of order kn which has the same optimal value. This opens new possibilities to get good lower bounds for several problems from combinatorial optimization, like the Quadratic Assignment Problem (QAP) and the Graph Partitioning Problem (GPP). In particular we show how to improve significantly the well-known Hoffman-Wielandt eigenvalue lower bound for QAP and the Donath-Hoffman eigenvalue lower bound for GPP by semidefinite programming. In the last part of the paper we show that the copositive strengthening of the semidefinite lower bounds for QAP and GPP yields the exact values.