  


An LMI description for the cone of Lorentzpositive maps
Roland Hildebrand (roland.hildebrandimag.fr) Abstract: Let $L_n$ be the $n$dimensional second order cone. A linear map from $\mathbb R^m$ to $\mathbb R^n$ is called positive if the image of $L_m$ under this map is contained in $L_n$. For any pair $(n,m)$ of dimensions, the set of positive maps forms a convex cone. We construct a linear matrix inequality (LMI) that describes this cone. Namely, we show that its dual cone, the cone of LorentzLorentz separable elements, is a section of some cone of positive semidefinite complex hermitian matrices. Therefore the cone of positive maps is a projection of a positive semidefinite matrix cone. The construction of the LMI is based on the spinor representations of the groups $\Spin_{1,n1}(\mathbb R)$, $\Spin_{1,m1}(\mathbb R)$. We also show that the positive cone is not hyperbolic for $\min(n,m) \geq 3$. Keywords: second order cone, hyperbolic polynomial, semidefinite cone Category 1: Linear, Cone and Semidefinite Programming (Semidefinite Programming ) Citation: Laboratory of Modelling and Calculus (LMC), University Joseph Fourier, Grenoble, France, December 2005 Download: [Postscript][PDF] Entry Submitted: 12/14/2005 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  