  We propose a primal-dual interior-point method (IPM) with convergence to
second-order stationary points (SOSPs) of nonlinear semidefinite optimization
problems, abbreviated as NSDPs. As far as we know, the current algorithms for
NSDPs only ensure convergence to first-order stationary points such as
Karush-Kuhn-Tucker points. The proposed method generates a sequence
approximating SOSPs while minimizing a primal-dual merit function for NSDPs by
using scaled gradient directions and directions of negative curvature. Under
some assumptions, the generated sequence accumulates at an SOSP with a
worst-case iteration complexity. This result is also obtained for a primal IPM
with slight modification. Finally, our numerical experiments show the benefits
of using directions of negative curvature in the proposed method.
