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 Semidefinite Programming) Relaxation Method and the SSILP (Successive Semi-Infinite Linear Programming) Relaxation Method, this paper proposes their implementable Variants for general quadratic optimization problems. These problems have a linear objective function c(T) x to be maximized over a nonconvex compact feasible region F described by a finite 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 infinite number of semi-infinite SDPs (or semi-infinite LPs) which appeared at each iteration of the original methods by a finite number of standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish: Given any opera convex set U containing F, there is an implementable discretization of rile SSDP (or SSILP) Relaxation Method which generates a compact convex set C such that F subset of or equal to C subset of or equal to U in a finite 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 Bred 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 ob redundant work to make the convex relaxation accurate in unnecessary directions. We establish: Given any positive number epsilon, there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method which generates art tipper bound of the objective value within epsilon of its maximum in a finite number of iterations.