We present a geometrical analysis on the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems with a class of geometrically defined nonconvex conic programs and their covexification. The class of nonconvex conic programs is described with a linear objective function in a linear space V, and the constraint set is represented geometrically as the intersection of a nonconvex cone K ⊂ V, a face J of the convex hull of K and a parallel translation L of a supporting hyperplane of the nonconvex cone K. We show that under a moderate assumption, 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 the hyperplane L. The replacement procedure is applied to derive the completely positive programming reformulation of quadratic optimization problems and its extension to polynomial optimization problems.
Key words.Completely positive reformulation of quadratic and polynomial optimization problems, conic optimization problems, hierarchies of copositivity, faces of the completely positive cone.AMS Classification. 90C20, 90C25, 90C26.widely studied subclass of POPs as they include many important NP-hard combinatorial problems such as binary QOPs, maximum stable set problems, graph partitioning problems and quadratic assignment problems. To numerically solve QOPs, a common approach is through solving their convex conic relaxations such as semidefinite programming relaxations [23,21] and doubly nonnegative (DNN) relaxations [15,19,26,28]. As those relaxations provide lower bounds of different qualities, the tightness of the lower bounds has been a very critical issue in assessing the strength of the relaxations. The completely positive programming (CPP) reformulation of QOPs, which provides their exact optimal values, has been extensively studied in theory. More specifically, QOPs over the standard simplex [9, 10], maximum stable set problems [12], graph partitioning problems [24], and quadratic assignment problems [25] are equivalently reformulated as CPPs. Burer's CPP reformulations [11] of a class of linearly constrained QOPs in nonnegative and binary variables provided a more general framework to study the specific problems mentioned above. See also the papers [1,2,8,14,22] for further developments.Despite a great deal of studies on the CPP relaxation, its geometrical aspects have not been well understood. The main purpose of this paper is to present and analyze essential features of the CPP reformulation of QOPs and its extension to POPs by investigating their geometry. With the geometrical analysis, many existing equivalent reformulations of QOPs and POPs can be considered in a unified manner and deriving effective numerical methods for computing tight bounds can be facilitated. In particular, the class of QOPs that can be equivalently reformulated as CPPs in our framework includes Burer's class of linearly constrained QOPs in nonnegative and...