Two Filter Algorithms For Nonlinear Semidefinite Programming

In this thesis,nonlinear semidefinite programming problems are studied with equality constraints and matrix inequality constraints.These problems are widely used in robust optimization problems,portfolio optimization prob-lems with marginal risk control in financial investment and truss problems in engineering design and so on.Thus,it is of great significance and value for the research on the algorithm of nonlinear semidefinite programming both in theory and in practical applications.Two new filter algorithms for solving nonlinear semidefinite program-ming are proposed in this thesis.Firstly,based on the idea of the filter method for traditional nonlinear programming,a filter algorithm for nonlinear semidef-inite programming is presented.In the algorithm,a penalty function is used as merit function to judge whether the trial point is accepted by the filter or not,but the penalty parameter is chosen as a constant and is not updated in the algorithm.When the subproblem of generating search direction is incom-patible or the constraint violation function value of the current iteration point rebounds too large,the algorithm needs to enter the feasibility restoration phase to generate a point closer to the feasible region.Under some mild as-sumptions,the global convergence of the proposed algorithm is proved.The numerical results show that the algorithm is effective.Secondly,since the restoration phase is relatively complex and will in-crease the computational cost of the algorithm,it will affect the overall ef-ficiency of the algorithm.Hence,based on the subproblem modified tech-niques of traditional nonlinear programming and combined with line search techniques,a new filter algorithm without feasibility restoration phase for nonlinear semidefinite programming is put forward.In this algorithm,the subproblem generating search direction is compatible,and the line search en-sures the descent of the constraint violation function,so the algorithm does not need the feasibility restoration phase.Under some conditions,the global convergence of the algorithm is proved.The numerical results show that the algorithm is effective.