A Geometrical Analysis on Convex Conic Reformulations of Quadratic and Polynomial Optimization Problems

Abstract

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...

Doubly nonnegative relaxations are equivalent to completely positive reformulations of quadratic optimization problems with block-clique graph structures

“…In this paper, we employ the following result, which is essentially equivalent to Burer's reformulation. See [17] for CPP reformulations of more general class of QOPs.…”

Sparse Conic Reformulation of Structured QCQPs based on Copositive Optimization with Applications in Stochastic Optimization

“…On closer inspection, it is not surprising that such a result can be obtained if one considers the identity between the cone of completable connected components and the set of completable partial set-completely positve matrices with arrowhead structure, to be discussed Section 6. However, our proof does not reference any type of completion result but is entirely based on geometrical arguments inspired by [16]. We hope that geometrical analysis of the cone of completable connected components will in the future allow for more general results on exact, reduced set-completely positive reformulations.…”

“…Thankfully, Kim et. al recently proposed a geometrical perspective on the proof strategy (see [16]). The concepts they introduce are quite versatile and allow proofs for generalizations of Theorem 1 as well exactness proofs for relaxations of polynomial optimization problems.…”

“…See [8] for the original proof for K = R n + , see [13] for the proof under the assumption that K is a norm-cone. Considering the results in [16], which we will discuss later in this text, no such assumption is needed for the theorem to hold.…”

An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs

“…Nevertheless, this unifying result led to intensive research activity along two main directions. On the one hand, Burer's result has been extended to larger classes of nonconvex optimization problems by introducing generalized notions of the copositive cone (see, e.g., [2][3][4]6,9,11,14,16,32], and also [24] for a recent geometric view of copositive reformulations). On the other hand, various tractable inner and outer approximations of the copositive cone have been proposed (see, e.g., [7,12,19,25,30,31,35]).…”