A Class Of Modified Trust Region Methods

Trust region method is an important numerical method developed over the last twodecades. Because of its reliability, robustness and strong convergence, trust region method,together with line search method, has become one of the two main numerical methods forsolving nonlinear programming. We mainly discuss the application of a modified BFGSformula to trust region method. The thesis is organized as follows:In Chapter 1, we review the basic thought and research of the trust region method.According to the new quasi Newton equation proposed by Wei at al., we give a new modifiedBFGS formula and analyze the corresponding properties.In Chapter 2, corresponding to the new formula, we develop a nonmonotone BFGStrust-region method for solving unconstrained optimization problems. A good property of themethod is that the objective function of the trust-region subproblem is a strictly convexquadratic function. Consequently, it is relatively easy to solve the subproblem. We establish aglobal convergence theorem for the method without requirement of the boundedness of thegenerated matrix sequence.In Chapter 3, combining the modified BFGS formula with Armijo line search, wepropose a new trust region method, and prove that the algorithms are global and superlinearconvergence under some mild conditions. Preliminary numerical results show that the methodis efficient and advisable for nonlinear optimization.