Several Methods For Unconstrained Optimizition Problems

The thesis mainly considers the convergence of line search methods and self-adaptive trust region methods for unconstrained optimizition problems. It comprises the following three chapters.Chapter 1 is the introduction, which mainly contains the research value and research situations of line search methods and self-adaptive trust region methods for unconstrained optimizition problems. In this chapter, we also discribe the main work based on the development.In chapter 2, the convergence of line search methods for unconstrained optimizition problems is proposed. We consider the unconstrained optimizition problem, Line search method is an effective method for unconstrained optimizition problems. Its convergence has attracted much more attention in recent years. Some new line search methods have been proposed year by year. In Ref. [12], the general results on convergence of line search methods with seven line search rules is proposed by Shi. These convergence results enable us to design powerful, effective, and stable algorithms in practice. Based on Ref. [12], we obtained the same convergence properties under weaker conditions than that of Ref. [12]. As a rule, some related algorithms can solve wider unconstrained optimizition problems.In chapter 3, a new self-adaptive trust region method and its convergence are proposed. In Ref. [24], an adaptive trust region method is proposed by zhang xiang sun etc., in which 0 < c < 1, and p is a positive integer, the local and global convergence results of the adaptive trust region method areproved. Motivated by the technique in Ref. [24], we adjust the trust region radius moderately , where A^ = cp "^—, 0 < c < 1, and p is a positive integer, In such a case,the method can avoid computing the inverse of the matrix in Ref. [24] and make the computation simple. We also proved the global convergence and local superlinear convergence rate of the new method under some mild conditions. The numerical results show that the new method is not only feasible but efficient, The results also show that the new adaptive trust region method with Bk modified by BFGS formula seems to be the best trust region method, This implies that the choice of Bk is very important for the new adaptive trust region method.