Interiors of completely positive cones

Abstract

ANWA ZHOU AND JINYAN FANÅ bstract. A symmetric matrix A is completely positive (CP) if there exists an entrywise nonnegative matrix B such that A " BB T . We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking interiors of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented. IntroductionA real nˆn symmetric matrix A is completely positive (CP) if there exist nonnegative vectors b 1 ,¨¨¨, b m P R n such that, where m is called the length of the decomposition (1.1). The smallest m in the above is called the CP-rank of A. If A is CP, we call (1.1) a CP-decomposition of A. Clearly, A is CP if and only if A " BB T for an entrywise nonnegative matrix B. Hence, a CP-matrix is not only positive semidefinite but also nonnegative entrywise.Let S n be the space of real nˆn symmetric matrices. For a cone C Ď S n , its dual cone is defined as C˚:" tG P S n : xA, Gy ě 0 for all A P Cu, where xA, Gy " tracepAGq is the trace inner product. Denote C n " tA P S n : A " BB T with B ě 0u, the completely positive cone, Cn " tG P S n : x T Gx ě 0 for all x ě 0u, the copositive cone.Both C n and Cn are proper cones (i.e. closed, pointed, convex and full-dimensional). Moreover, they are dual to each other [17]. The completely positive cone and copositive cone have wide applications in mixed binary quadratic programming [6], standard quadratic optimization problems and general quadratic programming [4], etc. Some NP-hard problems can also be formulated as linear optimization problems over the CP cone and the copositive cone (cf. [8]). We refer to [3,5,14] for the work in the field.The membership problems for the completely positive cone and the copositive cone are NP-hard (cf. [1,13]). To compute a CP-decomposition of a CP-matrix is also hard. Dickinson & Dür [9] studied the CP-checking and CP-decomposition of some sparse matrices. Sponseldur & Dür [28] used polyhedral approximations to project a matrix to C n ; a CP-decomposition of a matrix can be obtained if it 2000 Mathematics Subject Classification. Primary: 15A48, 65K05, 90C22, 90C26.

Building a completely positive factorization

“… 2015 ), to cite just a few. Another recent paper (Zhou and Fan 2015 ) deals with algorithmic strategies for factorization based upon conic optimization, for random instances of relatively moderate order .…”

A semidefinite algorithm for completely positive tensor decomposition

“…If A is completely positive, (1.1) is called a nonnegative (or completely positive) decomposition of A. The completely positive tensor is an extension of the completely positive matrix [1,2,33,34,35]. For B ∈ S m (R n ), we define Bx m := 1≤i1,...,im≤n B i1,...,im x i1 • • • x im .…”

The CP-Matrix Approximation Problem

“…By Nie's approach proposed in [27,28], Zhou and Fan [36] presented a semidefinite algorithm for the CP-matrix completion problem, which includes the CP checking as a special case; a CP-decomposition for a general CP-matrix can also be found by the algorithm. The approach is also applied to check interiors of the completely positive cone [37].…”