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


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


In this talk, we consider a primal-dual interior point method for nonlinear semidefinite programming problem:
min  & f(x), &  x ∈ {\bf R}n, \ 
s.t.  & g(x)=0, &  X(x) \succeq 0,
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.
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 l1- Penalty Interior Point Method

Coauthors: Javier Moguerza, Andrés Redchuk


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 l1-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


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