## 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.