Continuous Characterizations of the Maximum Clique Problem 1

Given a graph G whose adjacency matrix is A, the Motzkin-Strauss formulation of the Maximum-Clique Problem is the quadratic program maxfx T Axjx T e = 1; x 0g. It is well known that the global optimum value of this QP is (1 ? 1=!(G)), where !(G) is the clique number of G. Here, we characterize the following: 1) rst order optimality 2) second order optimality 3) local optimality 4) strict local optimality. These characterizations reveal interesting underlying discrete structures , and are polynomial time veriiable. A parametrization of the Motzkin-Strauss QP is then introduced and its properties are investigated. Finally, an extension of the Motzkin-Strauss formulation is provided for the weighted clique number of a graph.