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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.
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