Scalings In Trust Region Method

Trust region methods arc an important class of iterative methods for the solution of nonlinear optimization problems. Algorithms in this class have been proposed for the solution of systems of nonlinear cquations, nonlinear estimation problems, unconstrained and constrained optimization, nondifferentiable optimization, and large scale optimization.The development of trust region methods can be traced back to the work of Levcnbcrg and Marquardt on nonlinear least squares problems. In recent years, trust region methods attract many scholars' attention because of their strong global convergence results and stabilities.Recently, many algorithms about trust region methods devote to solve the subproblcm to improve the efficiency. From a theoretical point of view, the scaling matrices are a nuisance. It is much easier to set D_k = I since this simplifies the exposition of the results. However, we prefer a reliable region in which the quadratic model could have a good approximation to the objective function. So from a practical point of view, scalings are necessary and important. In our thesis, we propose a method to scale the region on the assumption that, the model contains more exact information about gradient and Hessian of the objective function. We present two kinds of scalings for unconstrained trust region methods, which are simple and explicit. Then some theoretical explanations on these methods and our experimental results are discussed.Moreover, we use the diagonal scaling matrix in box-constrained problems, and the experiment results show good.