Two Primal Dual Interior Point Algorithms For Convex Quadratic Semidefinite Programming

In this thesis,a special kind of nonlinear semidefinite programming,i.e.,convex quadratic semidefinite programming(CQSDP for short)is inves-tigated.CQSDP has a wide range of applications in the fields of economy,finance,engineering design,control theory and so on.Therefore,the study on algorithms for solving the CQSDP has important practical significance in theory and application.In this thesis,two primal dual interior point algorithms are proposed:one is primal dual potential reduction interior point algorithm;the other is long step primal dual path-following algorithm.Firstly,motivated from the idea of primal dual interior point method,based on primal dual affine-scaling direction and NT-scaling direction as well as a potential function,we propose a potential reduction interior point algorithm.The algorithm has the follow-ing properties:when the iterative point is near the center path,affine-scaling direction is used as the search direction which ensures that the potential func-tion is sufficiently reduced;when the iterative point is far from the center path,NT-scaling direction is taken which also guarantees that the potential function has sufficiently descent property;the algorithm will stop in at most O(?nln1/?)iterations.Secondly,based on the idea of long-step primal dual path-following method for linear semidefinite programming,by introducing a primal dual logarithmic barrier function,and using NT direction as a search direction,a long-step primal dual path-following algorithm for CQSDP is put forward.The algorithm has the following properties:the logarithmic barrier function is sufficiently reduced;the full step is accepted when the iterative point is near the center path;an ?-optimal is found in at most D(n|ln?|)iterations.Finally,some preliminary numerical results for the proposed algorithms are reported.The numerical results indicate that the proposed algorithms are feasible and effective.