My problem is to characterise (or find useful information on) the cone $C$ of $N\times N$ matrices $M$ ($N\geq 1$) such that $$V^t M V\geq 0$$ for every vector $V $ with non-negative entries. Is this cone of matrices familiar to anyone?


Remark 1: $C$ clearly contains the convex cone $S$ of semi-definite positive matrices, and that of matrices $P$ with non-negative entries. Even though it looks too easy, I could not prove that $C$ is not the convex hull of $S$ and $P$.


Remark 2: My original problem is to characterise the dual cone of $C$, containing the covariances of random vectors of $\mathbb{R}^N$ with a.s. non-negative entries.



Characterising semi-definite positiveness on vectors with non-negative entries
The cone $C$ is called the cone of copositive matrices and its dual $C^\*$ is called the cone of completely positive matrices. Here are some references.


The paper most relevant to your question is probably "On Non-Negative Forms In Real Variables Some Or All Of Which Are Non-Negative," in which P. H. Diananda shows that $C$ does in fact take the simple form you stated in Remark 1 in the case $n\leq 4$. As mentioned in a note at the end of that paper, a counterexample called the Horn Form shows that $C$ does not have this form when $n\geq 5$.


Another important paper in this area is "Some NP-Complete Problems In Quadratic And Nonlinear Programming" by K. G. Murty and S. N. Kabadi. There the authors show that checking membership in $C$ is co-NP-complete. In "Semidefinite programming based tests for matrix copositivity," P. A. Parrilo constructs a hierarchy of outer approximations to $C$ defined via SDPs; their duals give inner approximations to $C^\*$. By combining these ideas one can easily construct semidefinite program approximations to a variety NP-complete optimization problems.


