On copositive programming and standard quadratic optimization problems

A standard quadratic problem consists of finding global maximizers of a qua dratic form over the standard simplex. In this paper, the usual semidefinit e programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the posi tive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matri ces (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual a ffine-scaling directions. Furthermore, these approaches are combined with a n evolutionary dynamics algorithm which generates primal-feasible paths alo ng which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics ph ase.