Two Globally Convergent QP-Free Algorithms For Nonlinear Semidefinite Programming

In this thesis, nonlinear semidefinite programming (NLSDP for short) problems are investigated. These problems have wide applications in many fields, for instance, engineering, economy, optimal control, the optimal struc-ture optimization, truss design problem, etc. Therefore, researching on stable and efficient numerical algorithms for nonlinear semidefinite programming has important theoretical significance and applied value.In this thesis, two algorithms are proposed:one is a sequential system-s of linear equations (SSLE or QP-free for short) algorithm; the other is a QP-free algorithm without a penalty function or a filter. Firstly, based on the ideas of primal-dual interior point methods of traditional nonlinear program-ming and the techniques of systems of linear equations, combining with the techniques of inexact monotone line search and penalty parameter updating, a QP-free algorithm for nonlinear semidefinite programming is proposed. At each iteration, the search direction is yielded by solving two systems of lin-ear equations with the same coefficient matrix, exact penalty function is used as the merit function for line search. Under mild conditions, such as some matrix being full of column rank, the global convergence of the proposed algorithm is shown.Secondly, based on the ideas of filter methods of traditional nonlinear programming, we improve the line search of the first algorithm. Using the line search technique which ensures that the objective function or constrain-t violation function is sufficiently reduced, a QP-free algorithm without a penalty function or a filter for nonlinear semidefinite optimization is pro-posed. The numerical performance of the algorithm is improved to some extent due to the fact that the penalty function and the filter are not used. The proposed algorithm is shown to be globally convergent under some suitable conditions.Finally, we report some preliminary numerical results for two algorithms which are proposed in this thesis, and numerical results show the feasibility and effectiveness of the two proposed algorithms.