Peter J. C. Dickinson

On the computational complexity of membership problems for the completely positive cone and its dual

Copositive programming has become a useful tool in dealing with all sorts of optimisation problems. It has however been shown by Murty and Kabadi (Math. Program. 39(2):117–129, 1987) that the strong membership problem for the copositive cone, that is deciding whether or not a given matrix is in the copositive cone, is a co-NP-complete problem. From this it has long been assumed that this implies that the question of whether or not the strong membership problem for the dual of the copositive cone, the completely positive cone, is also an NP-hard problem. However, the technical details for this have not previously been looked at to confirm that this is true. In this paper it is proven that the strong membership problem for the completely positive cone is indeed NP-hard. Furthermore, it is shown that even the weak membership problems for both of these cones are NP-hard. We also present an alternative proof of the NP-hardness of the strong membership problem for the copositive cone.

AN IMPROVED CHARACTERISATION OF THE INTERIOR OF THE COMPLETELY POSITIVE CONE

A symmetric matrix is defined to be completely positive if it allows a factorisa- tion BB T , where B is an entrywise nonnegative matrix. This set is useful in certain optimisation problems. The interior of the completely positive cone has previously been characterised by Dur and Still (M. Dur and G. Still, Interior points of the completely positive cone, Electronic Journal of Linear Algebra, 17:48-53, 2008). In this paper, we introduce the concept of the set of zeros in the nonnegative orthant for a quadratic form, and use the properties of this set to give a more relaxed characterisation of the interior of the completely positive cone.

LINEAR-TIME COMPLETE POSITIVITY DETECTION AND DECOMPOSITION OF SPARSE MATRICES ∗

A matrix

On the exhaustivity of simplicial partitioning

X

On an extension of Pólya's Positivstellensatz

is called completely positive if it allows a factorization

A fresh CP look at mixed-binary QPs: new formulations and relaxations

X = \sum_{b\in \mathcal{B}} bb{^\mathsf{T}}

Moment Approximations for Set-Semidefinite Polynomials

with nonnegative vectors

Erratum to: On the DJL conjecture for order 6

b

Geometry of the copositive and completely positive cones

. These matrices are of interest in optimization, as it has been found that several combinatorial and quadratic problems can be formulated over the cone of completely positive matrices. The difficulty is that checking complete positivity is

The copositive cone, the completely positive cone and their generalisations

\mathcal{NP}