Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization

Based on the authors' previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semide nite Programming) Relaxation Method and the SSILP (Successive Semi-In nite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have a linear objective function cTx to be maximized over a nonconvex compact feasible region F described by a nite number of quadratic inequalities. We introduce two new techniques, \discretization" and \localization," into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an in nite number of semi-in nite SDPs (or semi-in nite LPs) which appeared at each iteration of the original methods by a nite number of standard SDPs (or standard LPs) with a nite number of linear inequality constraints. We establish: Given any open convex set U containing F , there is an implementable discretization of the SSDP (or SSILP) Relaxation Method which generates a compact convex set C such that F C U in a nite number of iterations. The localization technique is for the cases where we are only interested in upper bounds on the optimal objective value (for a xed objective function vector c) but not in a global approximation of the convex hull of F . This technique allows us to generate a convex relaxation of F that is accurate only in certain directions in a neighborhood of the objective direction c. This cuts o redundant work to make the convex relaxation accurate in unnecessary directions. We establish: Given any positive number , there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method which generates an upper bound of the objective value within of its maximum in a nite number of iterations.