A New Class Of Nonmonotone Trust Region Methods

Trust region method is a class of very important numerical algorithms for nonlinear optimization. Because of its strong convergence, robustness and validity, researchers in nonlinear optimization have paid great attention to it since 1970s. At present, trust region method and line search method are mainly two types of numerical algorithms for nonlinear optimization.Till now, after making great efforts, many researchers have proposed many kinds of trust region methods, such as quasi-Newton trust region method, trust region interior point method, nonmonotone trust region method, self-adaptive trust region method, and so on. A new kind of nonmonotone trust region methods which further enriches the research on trust region methods is presented in this thesis. The main content of the thesis is presented as follows:In chapter 1, we mainly review the basic idea, main theories and corresponding research results of trust region methods.In chapter 2, a nonmonotone trust region method based on the special weighted average is proposed. In the traditional nonmonotone trust region method, the reference function value of the iteration point is the maximum function value of a prefixed number of previous iterates. But in our method, the reference function value of the iteration point is defined by a special weighted average, i.e., a convex combination of all previous function values. Global and superlinear convergence of the method are proved under suitable conditions. Some numerical experiments show that the new method is superior to the traditional nonmonotone trust region methods on some problems.In chapter 3, on the base of the nonmonotone trust region method presented in chapter 2, a nonmonotone trust region method with fixed stepsize is proposed. "Unlike it, if the trial step is rejected, a stepsize is computed by a fixed formula along the trial step, which not only avoids resolving the subproblem repeatedly but also reduces the numbers of...