A GEOMETRICAL ANALYSIS ON CONVEX CONIC REFORMULATIONS OF QUADRATIC AND POLYNOMIAL OPTIMIZATION PROBLEMS

Abstract

We present a unified geometrical analysis on the completely positive programming (CPP) reformulations of quadratic optimization problems (QOPs) and their extension to polynomial optimization problems (POPs) based on a class of geometrically defined nonconvex conic programs and their convexification. The class of nonconvex conic programs minimize a linear objective function in a vector space V over the constraint set represented geometrically as the intersection of a nonconvex cone K subset of V, a face J of the convex hull of K, and a parallel translation L of a hyperplane. We show that under moderate assumptions, the original nonconvex conic program can equivalently be reformulated as a convex conic program by replacing the constraint set with the intersection of J and L. The replacement procedure is applied for deriving the CPP reformulations of QOPs and their extension to POPs.