- An LMI description for the cone of Lorentz-positive 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 Lorentz-Lorentz 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,n-1}(\mathbb R)\$, \$\Spin_{1,m-1}(\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 (Semi-definite Programming ) Citation: Laboratory of Modelling and Calculus (LMC), University Joseph Fourier, Grenoble, France, December 2005 Download: [Postscript][PDF]Entry Submitted: 12/14/2005Entry Accepted: 12/14/2005Entry Last Modified: 01/17/2006Modify/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 Optimization Online is supported by the Mathematical Programming Society and by the Optimization Technology Center.