  We study properties of the central path underlying a nonlinear semidefinite
optimization problem, called NSDP for short. The latest radical work on this
topic was contributed by Yamashita and Yabe (2012): they proved that the
Jacobian of a certain equation-system derived from the Karush-Kuhn-Tucker (KKT)
conditions of the NSDP is nonsingular at a KKT point under the second-order
sufficient condition (SOSC), the strict complementarity condition (SC), and the
nondegeneracy condition (NC). This yields uniqueness and existence of the
central path through the implicit function theorem. In this paper, we consider
the following three assumptions on a KKT point: the strong SOSC, the SC, and
the Mangasarian-Fromovitz constraint qualification. Under the absence of the
NC, the Lagrange multiplier set is not necessarily a singleton and the
nonsingularity of the above-mentioned Jacobian is no longer valid. Nonetheless,
we establish that the central path exists uniquely, and moreover prove that the
dual component of the path converges to the so-called analytic center of the
Lagrange multiplier set. As another notable result, we clarify a region around
the central path where Newton's equations relevant to primal-dual interior
point methods are uniquely solvable.
