Contributed Session Thu.2.H 2038

Thursday, 13:15 - 14:45 h, Room: H 2038

Cluster 4: Conic programming [...]

Interior-point methods for conic programming

 

Chair: Bo Kyung Choi

 

 

Thursday, 13:15 - 13:40 h, Room: H 2038, Talk 1

Chek Beng Chua
Weighted analytic centers for convex conic programming

 

Abstract:
We extend the target map, together with the weighted barriers and the weighted analytic centers, from linear programming to general convex conic programming. This extension is obtained from a novel geometrical perspective of the weighted barriers, that views a weighted barrier as a weighted sum of barriers for a strictly decreasing sequence of faces. Using the Euclidean Jordan-algebraic structure of symmetric cones, we give an algebraic characterization of a strictly decreasing sequence of its faces, and specialize this target map to produce a computationally-tractable target-following algorithm for symmetric cone programming. The analysis is made possible with the use of triangular automorphisms of the cone, a new tool in the study of symmetric cone programming.

 

 

Thursday, 13:45 - 14:10 h, Room: H 2038, Talk 2

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

 

Abstract:
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.

 

 

Thursday, 14:15 - 14:40 h, Room: H 2038, Talk 3

Bo Kyung Choi
New large-update primal-dual interior-point algorithms for symmetric optimization problems

Coauthor: Gue Myung Lee

 

Abstract:
A linear optimization problem over a symmetric cone, defined on a Euclidean Jordan algebra and called a symmetric optimization problem (shortly, SOP), is considered. We formulate an large-update primal-dual interior-point algorithm for SOP by using the proximity function defined by a new kernel function, and obtain complexity results for our algorithm by using the Euclidean Jordan algebra techniques.

 

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