On the central path for nonlinear semidefinite programming

In this paper we study the well definedness of the central path associated to a given nonlinear (convex) semidefinite programming problem. Under standard assumptions, we establish that the existence of the central path is equivalent to the nonemptiness and boundedness of the optimal set. Other equivalent conditions are given, such as the existence of a strictly dual feasible point or the existence of a single central point. The monotonic behavior of the logarithmic barrier and the objective function along the trajectory is also discussed. Finally, the existence and optimality of cluster points are established.