Error Bounds and Limiting Behavior of Weighted Paths Associated with the SDP Map X1/2SX1/2

This paper studies the limiting behavior of weighted infeasible central paths for semidefinite programming (SDP) obtained from centrality equations of the form X1/2SX1/2 = $\nu$ 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 $\sqrt{\nu}$ can be extended analytically beyond $0$ and hence that the path converges as $\nu \downarrow 0$. Characterization of the limit points of the path and its normalized first-order derivatives are also provided. As a consequence, it is shown that a weighted central path can have two types of behavior: it converges either as ${\Theta}(\nu)$ or as ${\Theta}(\sqrt{\nu})$ depending on whether the matrix W on a certain scaled space is block diagonal or not, respectively. We also 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. Finally, in light of the results of this paper, we give a characterization of a sufficient condition proposed by Potra and Sheng which guarantees the superlinear convergence of a class of primal-dual interior-point SDP algorithms.