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 XS + SX = 2νW , where W is a fixed positive definite matrix and ν > 0 is a parameter, under the assumption that the problem has a strictly complementary primal–dual optimal solution. We present a different and simpler proof than the one given by Preiß and Stoer [Preiß, M. and Stoer, J., 2004, Analysis of infeasible-interior-point paths arising with semidefinite linear complementarity problems. Mathematical Programming 99, 499–520.] that a weighted central path as a function of ν can be extended analytically beyond 0. In addition, the characterization of the limit points of the path and its normalized first-order derivatives is also provided. 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, we make some observation for the superlinear convergence of some primal–dual interior point SDP algorithms usingAHO neighborhood.