Some Researchs On Trust Region Method For Unconstrained Optimization Problems

In this thesis, we mainly discuss the trust region algorithms with line search for solving unconstrained optimization problems. At present, trust region and line search methods are two prominent classes of iterative methods to solve the uncon-strained optimization problems. Compared with line search method, trust region method needs less number of iterations and has robust and strong convergence. Be-sides, trust region method can solve not only good but also ill-conditioned problems. Researchers of the optimization field paid more attention to the trust region method in the past thirty and forty years. The traditional trust region method resolves the trust region subproblem when the trial step is not accepted, cause not easy to find new iteration, large amount of calculation and other shortcomings. To overcome the defects of resolving the trust region subproblems, researchers Jorge and Yua[6,7], Michael Gertz[8] proposed new type trust region algorithms that combining line search rules. In this thesis, two trust region methods with line search are proposed for solving unconstrained optimization problems, main content are as follows:In Chapter1, based on the traditional trust region method, a trust region algorithm with new line search is proposed for solving unconstrained optimization problems. The stepsize is obtained making use of larger Armijo line search rule. The proposed algorithm overcomes the shortcomings of large amount of calculation when solving the subproblem at each iteration, therefore, it is more attractive for large scale optimization problems. The global convergence of the algorithm is proved under suitable conditions. Some numerical results are reported, which shows that our algorithm is quite effective.In Chapter2, a new nonmonotone inexact line search rule is proposed and applied to the trust region method for unconstrained optimization problems. In our line search rule, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of the current objective function value. We can obtain a larger stepsize in each line search procedure and possess nonmonotonicity when incorporating the nonmonotone term into the trust region method. Unlike the traditional trust region method, the algorithm avoids resolving the subproblem if a trial step is not accepted. Under suitable conditions, global convergence is established. Numerical results show that the new method is effective for solving unconstrained optimization problems.In the last chapter, we make a conclusion of the work and look forward to the future research work.