## Contributed Session Tue.3.H 0107

#### Tuesday, 15:15 - 16:45 h, Room: H 0107

**Cluster 16: Nonlinear programming** [...]

### Interior-point methods

**Chair: Mouna Ali Hassan**

**Tuesday, 15:15 - 15:40 h, Room: H 0107, Talk 1**

**Li-Zhi Liao**

A study of the dual affine scaling continuous trajectories for linear programming

**Abstract:**

In this talk, a continuous method approach is adopted to study

both the entire process and the limiting behaviors of the dual

affine scaling continuous trajectories for linear linear

programming. Since the approach is different from any existing

one, many new theoretical results on the trajectories resulted

from the dual affine scaling continuous method model for

linear programming are obtained.

**Tuesday, 15:45 - 16:10 h, Room: H 0107, Talk 2**

**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.

* *

*
*

**Tuesday, 16:15 - 16:40 h, Room: H 0107, Talk 3**

**Mouna Ali Hassan**

The *l*_{1}- Penalty Interior Point Method

**Coauthors: Javier Moguerza, AndrĂ©s Redchuk**

**Abstract:**

The problem of general nonconvex, nonlinear constraint optimization is

addressed, without assuming regularity conditions on the constraints,

and the problem can be degenerate. We reformulate the problem by

applying *l*_{1}-exact penalty function with shift variables to relax and

regularize the problem. Then a feasible type line search primal-dual

interior point method, approximately solve a sequence of inequality

constraint penalty-barrier subproblems. To solve each subproblems, a

Cauchy step would be computed beside to Newton step and the proposed

algorithm would move along a direction in the span of these two

steps. The penalty parameter is checked at the end of each iteration

as we do with the barrier parameter, since we do not need to update

the penalty parameter before performing the line search. If the multipliers are finite, then the corresponding penalty parameter is finite. Global convergence properties do not require the regularity conditions on the original problem. The solution to the penalty-barrier problem converge to the optima that may satisfy the Karush-Kuhn-Tuker point or Fritz-John point, and may satisfy a first-order critical point for the measure of the