Global escape strategies for maximizing quadratic forms over a simplex

Consider the problem of maximizing a quadratic form over the standard simplex. Problems of this type occur, e.g., in the search for the maximum (weighted) clique in an undirected graph. In this paper, copositivity-based escape procedures from inefficient local solutions are rephrased into lower-dimensional subproblems which are again of the same type. As a result, an algorithm is obtained which tries to exploit favourable data constellations in a systematic way, and to avoid the worst-case behaviour of such NP-hard problems whenever possible. First results on finding large cliques in DIMACS benchmark graphs are encouraging.