Two Sequential Semidefinite Programming Algorithms Without A Penalty Function Or A Filter For Nonlinear Semidefinite Programming

In this thesis, nonlinear semidefinite programming problems are investi-gated, which have many important applications, such as engineering design, optimal structure design, optimal robust control and robust feedback control design. Therefore, it has very important theoretical and practical significance to study the numerical algorithms for nonlinear semidefinite programming.In this thesis, we propose two new sequential semidefinite program-ming (SSDP for short) algorithms without a penalty function or a filter to solve nonlinear semidefinite programming. Firstly, based on the idea of SQP method without a penalty function or a filter for traditional nonlinear pro-gramming, we present a new SSDP algorithm without a penalty function or a filter to solve nonlinear semidefinite programming. The algorithm has the following properties:the initial point is arbitrary; iteration points are not required to be feasible; it overcomes the difficulty of choosing suitable penalty parameter due to no using any penalty function; a filter is not intro-duced by setting the upper bound of the constraint violation function. Under the Mangasarian-Fromovitz constraint qualification and other appropriate as-sumptions, we prove that the algorithm possesses global convergence.Secondly, based on the ideas of modified SQP algorithm for traditional nonlinear programming problem with inequality constraints, we put forward a modified SSDP algorithm without a penalty function or a filter to solve non-linear semidefinite programming problem only with a matrix inequality con-straint. This algorithm guarantees the consistency of quadratic semi-definite programming (QSD for short) subproblems. At each iteration, the search di-rection is yielded by solving a linear semidefinite programming (LSDP for short) subproblem and a modified QSD subproblem. The nomontone line search technique is used, which is different from that of the previous algo-rithm. Consequently, the numerical performance is improved to some extent. Under some appropriate assumptions, global convergence of the proposed algorithm is shown.Some preliminary numerical experiments show that the feasibility and effectiveness of the two proposed algorithms.