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In this paper, we propose a primal-dual interior point trust-region method for solving nonlinear semidefinite programming problems, in which the iterates converge to a point that satisfies the first-order and second-order optimality conditions. The method consists of the outer iteration (SDPIP-revised) that finds a Karush-Kuhn-Tucker (KKT) point which satisfies the second-order optimality condition, and the inner iteration (SDPTR-revised) that calculates an approximate barrier KKT point. Algorithm SDPTR-revised uses a commutative class of Newton-like directions within the framework of the trust-region method in the primal-dual space. In addition, we also use a direction of negative curvature when it exists. The proposed algorithm employs a new method that generates negative-curvature directions in the existence of-type penalty term for equality constraints. It is proved that there exists a limit point of the generated sequence which satisfies the second-order optimality condition along with the barrier KKT conditions.

Convergence to a second-order critical point by a primal-dual interior point trust-region method for nonlinear semidefinite programming