**Friday, 11:00 - 11:25 h, Room: H 2038**

**Matsukawa Yasuaki**

A primal barrier function phase I algorithm for nonsymmetric conic optimization problems

**Coauthor: Yoshise Akiko**

**Abstract:**

We call the set of positive semidefinite matrices whose elements are nonnegative the doubly nonnegative (DNN) cone. The DNN cone can be represented as a projection of a symmetric cone given by the direct sum of the semidefinite cone and the nonnegative orthant. Using the symmetric cone representation, the authors demonstrated the efficiency of the DNN relaxation and showed that it gives significantly tight bounds for a class of quadratic assignment problems while the computational time is too long. The result suggests a primal barrier function approach for the DNN optimization problem. However, most of existing studies on the approach have assumed the availability of a feasible interior point which is not practical. Motivated by these observations, we propose a primal barrier function Phase I algorithm for solving conic optimization problem over the closed convex cone K such that (a) K is not necessarily symmetric, (b) a self-concordant function is defined over the interior of K, and (c) its dual cone is not explicit or is intractable, all of which are observed when K is the DNN cone. We analyze the algorithm and provide a sufficient condition for finite termination.

Talk 2 of the invited session Fri.1.H 2038

**"Recent developments of theory and applications in conic optimization II"** [...]

Cluster 4

**"Conic programming"** [...]