# Limiting behavior of the Alizadeh-Haeberly-Overton weighted paths in semidefinite programming

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming obtained from centrality equations of the form $X S + SX = 2 \nu W$, where $W$ is a fixed positive definite matrix and $\nu>0$ is a parameter, under the assumption that the problem has a strictly complementary primal-dual optimal solution. It is shown that a weighted central path as a function of $\nu$ can be extended analytically beyond $0$ and hence that the path and its derivatives converge as $\nu \downarrow 0$. Characterization of the limit points and first-order derivatives of the scaled weighted central path is also provided. We finally derive an error bound on the distance between a point lying in a certain neighborhood of the central path and the set of primal-dual optimal solutions.

## Citation

Manuscript, School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, July 2003.