The CP-Matrix Approximation Problem

Abstract

A symmetric matrix A is completely positive (CP) if there exists an entrywise nonnegative matrix V such that A = V V T . In this paper, we study the CP-matrix approximation problem of projecting a matrix onto the intersection of a set of linear constraints and the cone of CP matrices. We formulate the problem as the linear optimization with the norm cone and the cone of moments. A semidefinite algorithm is presented for the problem. A CP-decomposition of the projection matrix can also be obtained if the problem is feasible.Both CP n and COP n are proper cones (i.e. closed, pointed, convex and fulldimensional). Moreover, they are dual to each other [17]. A variety of NP-hard problems can be formulated as optimization problems over the completely positive cone or the copositive cone. Interested readers are referred to [2,6,7,8,4,12,15] for the work in the field.The important applications of the CP cone motivate people to study whether a matrix is CP or not. However, checking the membership in CP n has been shown NPhard, while checking the membership in COP n co-NP-hard [13,26]. It is generally difficult to treat CP n (or COP n ) directly. A standard approach is to approximate 2000 Mathematics Subject Classification. Primary: 90C20, 90C22, 90C26.

Dehomogenization for completely positive tensors

“…The dehomogenization map can be used to characterize the CP moment cone CP n,d . Recall the truncated moment cone R d (∆) as in (10).…”

“…(ii) Consider the symmetric tensor A ∈ S 6 (R 4 ) such that φ(A) = (3,3,4,3,3,4,3,6,4,5,3,4,3,6,4,5,10,6,6,9,3,4,3,6,4,5,10,6,6,9,18,10,8,10,17,3,4,3,6,4,5,10,6,6,9,18,10,8,10,17,34,18,12,…”

“…The optimization (49) is called the CP tensor approximation problem. For the special case that d = 2, it is reduced to be a CP-matrix approximation problem (see the work [10,36]).…”

“…The CP tensors can be used to construct many interesting problems. Recent work for CP matrix approximation problem is given in [10,36]. Copositive tensors and CP tensors also have various applications.…”

A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

“…We also discuss the best CP tensor approximation problem, which is to find a tensor in the intersection of a set of linear constraints and the cone of CP tensors such that it is close to a given tensor as much as possible. It is an extension of the CP-matrix approximation problem [10]. We transform the problem to a conic linear program over the cone of moments and the second-order cone.…”