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Relating Homogeneous Cones and Positive Definite Cones via $T$-algebras

Chek Beng Chua (cbchua***at***math.uwaterloo.ca)

Abstract: $T$-algebras are non-associative algebras defined by Vinberg in the early 1960's for the purpose of studying homogeneous cones. Vinberg defined a cone $K(\mathcal A)$ for each $T$-algebra $\mathcal A$ and proved that every homogeneous cone is isomorphic to one such $K(\mathcal A)$. We relate each $T$-algebra $\mathcal A$ with a space of linear operators in such a way that $K(\mathcal A)$ is isomorphic to the cone of positive definite self-adjoint operators. Together with Vinberg's result, we conclude that every homogeneous cone is isomorphic to a ``slice'' of a cone of positive definite matrices.

Keywords: homogeneous cones, T-algebras, semidefinite programming, interior-point methods

Category 1: Linear, Cone and Semidefinite Programming (Other )

Category 2: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: SIAM J. Optim., 2003, 14:500-506

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Entry Submitted: 05/06/2002
Entry Accepted: 05/07/2002
Entry Last Modified: 03/29/2004

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