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Roland Hildebrand(roland.hildebrandimag.fr) Abstract: On the interior of a regular convex cone $K \subset \mathbb R^n$ there exist two canonical Hessian metrics, the one generated by the logarithm of the characteristic function, and the ChengYau metric. The former is associated with a selfconcordant logarithmically homogeneous barrier on $K$ with parameter of order $O(n)$, the universal barrier. This barrier is invariant with respect to the unimodular automorphism subgroup of $K$, is compatible with the operation of taking product cones, but in general it does not behave well under duality. In this contribution we introduce the barrier associated with the ChengYau metric, the EinsteinHessian barrier. It shares with the universal barrier the invariance, existence and uniqueness properties, is compatible with the operation of taking product cones, but in addition is invariant under duality. The EinsteinHessian barrier can be characterized as the convex solution of the partial differential equation $\log\det F'' = 2F$ with boundary condition $F_{\partial K} = +\infty$. Its barrier parameter does not exceed the dimension $n$ of the cone. On homogeneous cones both barriers essentially coincide. Keywords: interiorpoint methods, selfconcordant barriers, convex cones Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Category 2: Linear, Cone and Semidefinite Programming (Other ) Citation: Download: [PDF] Entry Submitted: 05/21/2012 Modify/Update this entry  
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