15th EUROPT Workshop on Advances in Continuous Optimization
Montréal, Canada, 12 — 14 juillet 2017
15th EUROPT Workshop on Advances in Continuous Optimization
Montréal, Canada, 12 — 14 juillet 2017
Copositive and completely positive problems
12 juil. 2017 09h45 – 11h00
Salle: Nancy et MichelGaucher
Présidée par Immanuel Bomze
3 présentations

09h45  10h10
New upper bounds on the kissing number via copositive programming
This paper introduces a hierarchy of upper bounds on the kissing number using copositive programming. Recently, it has been shown that the kissing number can be obtained from an infinite dimensional optimization problem over copositive kernels on a sphere. To construct a new semidefinite hierarchy for the kissing number, we extend an existing sdpbased hierarchy for the finite dimensional copositive cone to the infinite dimensional case and exploit symmetry of the sphere. Also, an alternative proof is given to characterize positive definite kernels invariant under automorphisms of the sphere with a given set of fixed points via Jacobi polynomials.

10h10  10h35
SPN completable graphs
An SPN matrix is a matrix which is the Sum of a Positive semidefinite and a symmetric Nonnegative one.
We consider the SPN completion problem: Given a partial matrix all of whose fully specified principal submatrices are SPN, can it be completed to an SPN matrix?
We characterize all the patterns of specified entries (given by a graph) which guarantee existence of such a completion.
Our result complements known characterizations of PSD, DNN, CP and COPcompletable graphs.
We also show how the characterization of completely positive graphs can be derived from our SPN completion result. 
10h35  11h00
The structure of completely positive matrices according to their CPrank and CPplusrank
Important in the study of completely positive matrices is the cprank, and here we introduce the closely related cpplusrank. Due to the subtle similarities and differences between the cpplusrank and the cprank, the analysis of the cpplusrank is highly useful in the investigation of the completely postive cone. We show numerous topological properties related to the cprank and cpplusrank, including that generically the cpplusrank is equal the cprank, and that in the interior of the completely positive cone the set of matrices whose cprank and cpplusrank both equal a fixed number is an open set.