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

Zhaosong Lu (zhaosongisye.gatech.edu)
Renato D. C. Monteiro (monteiroisye.gatech.edu)

Abstract: 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.

Keywords: limiting behavior, weighted central path, error bound,semidefinite programming.

Category 1: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

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

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Entry Submitted: 07/23/2003
Entry Accepted: 07/23/2003
Entry Last Modified: 12/12/2004

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