In this paper we analyze the limiting behavior of infeasible weighted central paths in semidefinite programming under the assumption that a strictly complementary solution exists. We show that the paths associated with the "square root" symmetrization of the weighted centrality condition are analytic functions of the barrier parameter $\mu$ even at $\mu=0$ if and only if the weight matrix is block diagonal in terms of optimal block partition of variables. This result strengthens some recent result by Lu and Monteiro establishing the analyticity of the paths as functions of $\sqrt\mu$ at $\mu=0$. Moreover, in this paper we study the analytical properties of the paths associated with the "Cholesky factor" symmetrization. We show that the paths exhibit the same analytical behavior at $\mu=0$ as the paths corresponding to the square root symmetrization.
Faculty of Mathematics, Physics and Informatics, Comenius University Bratislava, December 2007