On the superlinear local convergence of a penalty-free method for nonlinear semidefinite programming

This paper is concerned with a sequentially semidefinite programming (SSDP) algorithm for solving nonlinear semidefinite programming problems (NLSDP), which does not use a penalty function or a filter. This method, inspired by the classic SQP method, calculates a trial step by a quadratic semidefinite programming subproblem at each iteration. The trial step is determined such that either the value of the objective function or the measure of constraint violation is sufficiently reduced. In order to guarantee global convergence, the measure of constraint violation in each iteration is required not to exceed a progressively decreasing limit. We prove the global convergence properties of the algorithm under mild assumptions. We also analyze the local behaviour of the proposed method while using a second order correction strategy to avoid Maratos effect. We prove that, under the strict complementarity and the strong second order sufficient conditions with the sigma term, the rate of local convergence is superlinear. Finally, some numerical results with nonlinear semidefinite programming formulation of control design problem with the data contained in COMPleibCOMPleib are given.