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Einstein-Hessian barriers on convex cones

Roland Hildebrand(roland.hildebrand***at***imag.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 Cheng-Yau metric. The former is associated with a self-concordant 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 Cheng-Yau metric, the Einstein-Hessian 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 Einstein-Hessian 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: interior-point methods, self-concordant barriers, convex cones

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Linear, Cone and Semidefinite Programming (Other )

Citation:

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Entry Submitted: 05/21/2012
Entry Accepted: 05/21/2012
Entry Last Modified: 05/21/2012

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