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We show the cp-rank of an integer doubly nonnegative 2 × 2 matrix does not exceed 11.

It is shown that, if for some m , 2 ≤ m < n all principal submatrices of an n -by- n P -matrix A are monotone, then A is monotone, generalizing a known result that also assumed symmetry. It is also shown that the P -matrix assumption cannot be entirely eliminated. However, in addition, A is doubly monotone (n = m − 1 above, without the P -matrix assumption) if and only if A − 1 is invertible path product. It was known that inverse M -matrices are path product (they are more than doubly monotone). But (invertible) path product matrices are more general, and this is the first full and functional characterization of them. Several examples are given to illustrate the results and their generality. [ABSTRACT FROM AUTHOR]

An SPN matrix is a matrix which is the sum of a real positive semidefinite matrix and a symmetric nonnegative one. We solve the SPN completion problem: we show that the SPN completable graphs are the graphs in which every odd cycle induces a complete subgraph. [ABSTRACT FROM AUTHOR]

The set of polynomials that are nonnegative over a subset of the nonnegative orthant (we call them set-semidefinite) have many uses in optimization. A common example of this type set is the set of copositive matrices, where we are effectively considering nonnegativity over the entire nonnegative orthant and are restricted to homogeneous polynomials of degree two. Lasserre (SIAM J. Optim., 21(3):864-885, ) has previously considered a method using moments in order to provide an outer approximation to this set, for nonnegativity over a general subset of the real space. In this paper, we shall show that, in the special case of considering nonnegativity over a subset of the nonnegative orthant, we can provide a new outer approximation hierarchy. This is based on restricting moment matrices to be completely positive, and it is at least as good as Lasserre's method. This can then be relaxed to give tractable approximations that are still at least as good as Lasserre's method. In doing this, we also provide interesting new insights into the use of moments in constructing these approximations. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we discuss the complete positivity of house matrices, i.e., doubly nonnegative matrices whose graph is a cycle with a chord between vertices 1 and 3. We describe the set of all possible supports of the columns of a nonnegative matrix B,such that; show that the cp-rank of a completely positive house matrix is at least ; give a complete characterization of the singular completely positive house matrices; show that their cp-rank is or n and characterize each case. [Copyright &y& Elsevier]

Abstract: In their paper “ Completely positive matrices”, Berman and Xu (2004) attempt to characterize which doubly nonnegative matrices are also completely positive. Most of the analysis in concerns a doubly nonnegative matrix A that has at least one off-diagonal zero component. To handle the case where A is componentwise strictly positive, Berman and Xu utilize an “edge-deletion” transformation of A that results in a matrix having an off-diagonal zero. Berman and Xu claim that A is completely positive if and only if there is such an edge-deleted matrix that is also completely positive. We show that this claim is false. We also show that two conjectures made in regarding completely positive matrices are both false. [Copyright &y& Elsevier]

Abstract: The convex cone of completely positive (CP) matrices and its dual cone of copositive matrices arise in several areas of applied mathematics, including optimization. Every CP matrix is doubly nonnegative (DNN), i.e., positive semidefinite and component-wise nonnegative, and it is known that, for only, every DNN matrix is CP. In this paper, we investigate the difference between DNN and CP matrices. Defining a bad matrix to be one which is DNN but not CP, we: (i) design a finite procedure to decompose any DNN matrix into the sum of a CP matrix and a bad matrix, which itself cannot be further decomposed; (ii) show that every bad DNN matrix is the sum of a CP matrix and a single bad extreme matrix; and (iii) demonstrate how to separate bad extreme matrices from the cone of CP matrices. [Copyright &y& Elsevier]

Abstract: Considered are continuous, positive Hadamard powers of entry-wise positive (nonnegative) matrices. Those that are eventually (in the sense of all Hadamard powers beyond some point) totally positive, totally nonnegative, doubly nonnegative and doubly positive are characterized. For example, for matrices with at least four rows and four columns, Hadamard powers greater than one of totally positive matrices need not be totally positive, but they are eventually totally positive. [Copyright &y& Elsevier]

Let Φk be the maximal cp-rank of all rank k completely positive matrices. We prove that Φk=k(k+1)/2−1 for k&ges;2. In particular we furnish a procedure to produce, for k&ges;2, completely positive matrices with rank k and cp-rank k(k+1)/2−1. [Copyright &y& Elsevier]