Broyden's Class Of Modified Quasi-Newton Methods And Their Applications

Optimization has a wide range of practical background in economics, management, engineering and so on. It is also closely related to the numerical methods in computational mathematics such as the differential equation and the nonlinear equations. Due to the efficient numerical performance and the fast theoretic convergence property, the Broyden's class of quasi-Newton methods have become welcome numerical methods for solving optimization problems. Since early 1960s, much attention has been paid to the theory of Broyden's class of quasi-Newton methods. There has taken good progress in the study of global and superlinear convergence properties for these methods. In particular, the global and local superlinear convergence of Broyden's class of quasi-Newton methods for convex minimizations have been well established.On the other hand, however, the study on the global behavior of Broyden's class of quasi-Newton methods for nonconvex problem is relatively fewer. Recently, Dai constructed a counterexample to show the nonconvergence of BFGS method with Wolfe line search if the objective is not convex. As many practical problems arising from engineering are nonconvex, it is important to develop quasi-Newton methods that are globally and superlineraly convergent for nonconvex minimization. The purpose of this thesis is to study this topic.In this thesis, we first develop a modified BFGS and a cautious BFGS method and show their global and superlinear convergence for nonconvex minimizations. We then propose a structured quasi-Newton method for solving the nonlinear least squares problems. At last, we propose a norm descent quasi-Newton method for solving a nonsmooth equation reformulation of the symmetric variational inequality problem. Under appropriate conditions, we establish its global and superlinear convergence.First, we consider the following unconstrained optimization problemminf(x), x∈R~n,where f : R~n—>R is a smooth function. Recently, aiming at enhancing the efficiency of quasi-Newton methods, Zhang, Deng and Chen, Wei, Yu and Yuan et al. proposed some new quasi-Newton methods based on some new secant equations. Their numerical experiments show that these methods do have some advantages. They also established the local superlinear convergence of the methods. However,...