New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.Comment: 8

Linear copositive programming: strong dual formulations and their properties

In Copositive Programming, a cost function is optimized over a cone of matrices that are positive semidefinite in the non-negative ortant. Being a fairly new field of research, Copositive Programming has already gained popularity. Duality theory is a rich and powerful area of convex optimization, which is central to understanding sensitivity analysis and infeasibility issues as well as to development of numerical methods. In this paper, we continue our recent research on dual formulations for linear Copositive Programming. The dual problems obtained in the paper satisfy the strong duality relations and do not require any additional regularity assumptions such as constraint qualifications. Different dual formulations have their own special properties, the corresponding feasible sets are described in different ways, so they can have an independent application in practice.publishe