For Solving Constrained Optimization Problems For A Class Of Adaptive Trust Region Method With Line Search

The trust region method is a kind of efficient and robust method to solve the general unconstrained optimization and its special situation, the nonlinear least squares problems. Because of its remarkable numerical reliability in conjunction with a sound and complete convergence theory, researchers in nonlinear optimization area have paid great attention to it since 80's. At present, trust region methods and line search methods are two mainly types numerical algorithms for nonlinear programming. The choice of the trust region radius has an important effect on the efficiency of the trust region methods. Hei [1] proposed a self-adaptive trust region algorithm, in which the trust region radius is updated at a variable rate by a R-function. Zhang et.al. [2] 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 methods. In 1991, Jorge Nocedal and Yaxiang Yuan first proposed a new algorithm [3] by employing both trust region and line searches and they gave an algorithm which combing trust region and line search for unconstrained optimization. The first part of the thesis proposes a self-adaptive trust region algorithm with line search to solve the general unconstrained optimization problem, in which we combin the self-adaptive trust region method in [1]with the traditional line search. We establish the global convergence result of the new method. Numerical results show that the new method is more efficient than the method in [1]. The second part of the thesis proposed a new nonmonotonic self-adaptive trust region method, to solve the general unconstrained optimization problem, in which we combin the self-adaptive trust region method in [1]with the nonmonotonic techniques. Since the first part of the thesis becomes a special case of the second part of the thesis.In chapter 1, we introduce the development of optimization and some extensive optimality conditions to determine the optimum solution. We review severalextensive line-search methods for unconstrained optimization .In chapter 2, we first introduce the trust region method simply. Several important developments of the trust region method are reviewed and some simple anlysis is given .In chapter 3, we propose a self-adaptive trust region algorithm with line search to solve the general unconstrained optimization problem, in which we combin the self-adaptive trust region method in [1]with the Wolfe line search. Under some assumption the convergence of the new method is proved.In chapter 4, we put the nonmonotonic techniques into the self-adaptive trust region method and propose a nonmonotonic self-adaptive trust region with line search. Under some assumption the convergence of the new method is proved.