Thursday, 13:45 - 14:10 h, Room: H 2038


Roland Hildebrand
A barrier on convex cones with parameter equal to the dimension


Self-concordant barriers are central in interior-point methods for conic programming. The speed of
interior-point methods based on a particular barrier depends on a scalar parameter, the barrier
parameter. Nesterov and Nemirovski showed that the universal barrier, which exists and is unique for
every regular convex cone, has a barrier parameter of order O(n), where n is the
dimension of the cone. We present another barrier, the Einstein-Hessian barrier, which also exists
and is unique for every regular convex cone, but has barrier parameter equal to n. In addition to
compatibility with taking product cones and invariance with respect to unimodular automorphisms of
the cone, which it shares with the universal barrier, the Einstein-Hessian barrier is also
compatible with duality. The level surfaces of the Einstein-Hessian barrier are characterized by the
property of being affine hyperspheres, objects well-known in differential geometry. We give also
another, more intuitive geometric characterization of these level surfaces. They are minimal
surfaces in the product of the primal and dual projective spaces associated to the ambient real
spaces where the cone and its dual reside.


Talk 2 of the contributed session Thu.2.H 2038
"Interior-point methods for conic programming" [...]
Cluster 4
"Conic programming" [...]


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