**Tuesday, 15:45 - 16:10 h, Room: H 0107**

**Atsushi Kato**

An interior point method with a primal-dual quadratic barrier penalty function for nonlinear semidefinite programming

**Coauthors: Hiroshi Yabe, Hiroshi Yamashita**

**Abstract:**

In this talk, we consider a primal-dual interior point method for nonlinear semidefinite programming problem:

\begin{eqnarray}

{

\begin{array}{lll}

min* & f(x), & x ∈ {\bf R}*^{n}, \

s.t.* & g(x)=0, & X(x) \succeq 0,*

\end{array}

.

\nonumber

\end{eqnarray}

where functions *f:{\bf R}*^{n} → {\bf R}, *g:{\bf R}*^{n} → {\bf R}^{m} and *X:{\bf R}*^{n} → {\bf S}^{p}

are sufficiently smooth, and *{\bf S}*^{p} denotes the set of *p*-th order real symmetric matrices.

\par

Our method is consists of the outer iteration (SDPIP) and the inner iteration (SDPLS).

Algorithm SDPIP finds a KKT point.

Algorithm SDPLS also finds an approximate shifted barrier KKT point. Specifically, we apply the Newton method to the shifted barrier KKT conditions. To globarize the method, we propose a differentiable merit function in the primal-dual space within the framework of line search strategy. We show its global convergence property.

* *

*
*Talk 2 of the contributed session Tue.3.H 0107

**"Interior-point methods"** [...]

Cluster 16

**"Nonlinear programming"** [...]