On the Central Paths and Cauchy Trajectories in Semidefinite Programming

In this work, we study the properties of central paths, defined with respect to a large class of penalty and barrier functions, for convex semidefinite programs. The type of programs studied here is characterized by the minimization of a smooth and convex objective function subject to a linear matrix inequality constraint. So, it is a particular case of convex programming with conic constraints. The studied class of functions consists of spectrally defined functions induced by penalty or barrier maps defined over the real nonnegative numbers. We prove the convergence of the (primal, dual and primal-dual) central path toward a (primal, dual, primal-dual, respectively) solution of our problem. Finally, we prove the global existence of Cauchy trajectories in our context and we recall its relation with primal central path when linear semidefinite programs are considered. Some illustrative examples are shown at the end of this paper.