On Conic Trust Region Methods For Solving Unconstrained And Bound Constrained Optimization Problems

Nonlinear optimization is a cross branch of computational mathematics and op-erations research. It has a wide range of applications in national defence, economy, finance, engineering, management and many other fields. Many scientific and engi-neering problems, such as assimilation problem in atmospheric science, protein fold-ing problem in life science, pattern recognition problem in information science, in-verse problem in earth science and so on, usually are large scale and highly nonlinear. So we urgently need to develop some efficient numerical methods for solving nonlin-ear optimization problem.Trust region method is a very effective method for solving nonlinear optimization problem. Conic trust region method is a generalization method of basic trust region method. In this thesis, we study the conic model trust region optimization method.First, we discuss a nonmonotone retrospective conic trust region method for un-constrained optimization problem. Compared with the traditional trust region method, the subproblem in the presented method is conic model. Meanwhile, we use non-monotone technique to speed up the convergence rate. The global convergence of the new algorithm is established under some mild conditions. Numerical experiments are given to show the validity of our algorithm.Next, a nonmonotone adaptive trust region method for unconstrained optimiza-tion problem, based on simple conic model, is proposed. We use a scalar matrix to replace the Hessian matrix or its approximation. The new algorithm is simple, and is easy to carry out. Moreover, it needs less memory capacitance and has lower compu-tational complexity. Under some conditions, we give the global and local convergence of the algorithm. Numerical results show that the new algorithm is effective for large scale unconstrained optimization problems.Inspired by the idea of affine scaling trust region method, we present a conic affine scaling trust region method for bound constrained optimization problem, which is an extension of affine scaling trust region method based on quadratic model. We give the descent bound of the model function in the new algorithm, and prove the global convergence of the new algorithm under the framework of trust region method. Numerical results are given to show the efficiency of the algorithm.At the end of this thesis, a combined method for bound constrained optimization problem, based on nonmonotone conic affine scaling trust region and line search, is presented. When the trial point can not be accepted by trust region, a new iterate point produced by line search technique is adopted, that reduces the cost of computation to some extent. Global convergence of the new method is shown to be guaranteed under some conditions. Numerical results are reported to show the performance of the new algorithm.