A RFTR Algorithm For Unconstrained Optimization

Trust Region Method is one of the effective methods for unconstrained optimiza-tion. The Filtering techniques which are used in constrained optimization can improve the efficiency of the algorithm.Recently, the retrospective idea is proposed, which can give a good estimation of trust region radius. Therefore, the research with the advantages of the above algorithms is of great theoretical significance and application value.In this paper, we propose a retrospective filter trust region algorithm for un-constrained optimization, which is based on the framework of the retrospective trust region method and associated with the technique of the multi-dimensional filter. The new algorithm gives a good estimation of trust region radius, relaxes the condition of accepting a trial step for the usual trust region methods. Under reasonable assump-tions, we analyze the first and second convergence of the new method and report the preliminary results of numerical tests. Meanwhile, we compare the results with those of the basic trust region algorithm, the filter trust region algorithm and the retrospective trust region algorithm, which shows the efficiency of the new algorithm.Our algorithm looks like a self-adaptive method based on the trust-region frame-work. However, our algorithm is not like the other self-adaptive methods which need to compute the gradient value and function value at the auxiliary point,but may measure the acceptance of the previous iterate and the current iterate for the new and old model function,respectively, which keep the robustness property of the trust-region method and further improve the efficiency of the algorithm.