Augmented Lagrangian Method For Nonconvex Semidefinite Programming And Mond-Weir Duality Theory For Multiobjective Semidefinite Programming

Nonconvex semidefinite programming is widely used in the field of control theory, sen-sitivity analysis, systems engineering and electronic engineering. In recent years, the penalty function method, Languagian method and various smoothing algorithms have been success-fully applied in solving the nonconvex semidefinite programming problems. Multiobjective semidefinite programming is a multiobjective optimization with semidefinite constraints. Re-search shows that it has a wide range of applications in engineering, management science and economic field. In this thesis, It is mainly considered the Mond-Weir duality theory for mul-tiobjectiv semidefinite programming and the augmented Lagrangian method for nonconvex semidefinite programming..The first part of the thesis concerns augmented Lagrangian method for nonconvex semidefinite programming problems. And the second part considers the Mond Weir duality for the multiobjective semidefinite programming problems.Specifically, it is established the equivalent relationship between the nonconvex semidefi-nite programming problems with negative semidefinite constraints and the nonconvex semidef-inite programming problems with equality constraints under the second order sufficient condi-tion and KKT optimality condition assumptions. Then, the nonconvex semidefinite program-ming problems with equivalent constraints are transformed into equivalent nonlinear program-ming problems with the means of straightened algorithm. An augmented Lagrange function is proposed for this nonconvex semidefinite programming without the strict complementary condition. Finally, the relationship between local optimality condition of the original problem and the local optimality of the corresponding augmented Lagrange function is established.As for multiobjective semidefinite programming problems, firstly, it is introduced classes of weakly strictly pseudo quasi type I function, strongly pseudo quasi type I function, weak-ly quasi strictly pseudo type I function and weak strictly pseudo type I function. Secondly, a Mond-Weir dual model for the multiobjective semidefinite programming problem is estab- lished, and the weak duality theorem and the strong duality theorem are given and proved. Finally, three types of optimal sufficient conditions for multiobjective semidefinite program-ing are given in the sense of I type function.