Adaptive Trust Region Algorithms For Unconstrained Optimization And Nonlinear Least Squares Problems

Trust region method is a kind of efficient methods to solve the general unconstrained optimization and its special situation, the nonlinear least squares problems. The choice of the trust region radius has an important effect on the efficiency of the trust region method. Hei [12] proposed a self-adaptive trust region algorithm, in which the trust region radius is updated at a variable rate by R-function. Zhang et. al. [26] also proposed an adaptive trust region method, in which the trust region radius depends on the gradient and Hessian of the current iterate point. Numerical results show that both the methods are more efficient than the traditional trust region method. The first part of this thesis proposes a new nonmonotonic self-adaptive trust region algorithm with line search to solve the general unconstrained optimization problem, in which we combine the self-adaptive trust region method in [12] with the popular nonmonotone technique and use combing trust region with line search techniques in [16]. We establish the global convergence result of the new method. Numerical results show that the new nonmonotonic method can save more computation than the method in [12]. The second part of this thesis puts the adaptive trust region method in [26] to [10] to propose an adaptive conic trust region method for nonlinear least squares problems, in which we can use inexact methods to solve the trust region subproblem approximately. The global and superlinear convergence results of the new algorithm are established. Numerical results show that the method is efficient.