The relationships among the central path in the context of semidefinite programming, generalized proximal-point method and Cauchy trajectory in a Riemannian manifolds is studied in this paper. First, it is proved that the central path associated to a general function is well defined. The convergence and characterization of its limit point is established for functions satisfying a certain continuity property. Also, the generalized proximal-point method is considered and it is proved that the correspondingly generated sequence is contained in the central path. As a consequence, both converge to the same point. Finally, it is proved that the central path coincides with the Cauchy trajectory in a Riemannian manifold.
Iusem, A.N., Svaiter, B.F., da Cruz Neto, J.X.: Central paths, generalized proximal point methods and Cauchy trajectories in Riemannian manifolds. SIAM J. Control Optim. 37(2), 566–588 (1999)
Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)