Several Non-monotone Trust Region Methods For Solving Unconstrained Optimization Problems

Trust region method is a kind of effective calculation method and plays an important role in solving optimization problems. In recent years, with the introduction of non-monotone technique, non-monotone trust region method has been highly valued in the field of optimization. Specifically, non-monotone technology can not only increases the possibility of finding the global optimal solution, but also allows faster convergence rate.Compared with the traditional trust region method, the existing non-monotone trust region methods have been improved greatly. However, when dealing with unconstrained optimization problems, those methods still facing the problem of solving the sub-problems repeatedly. And, they are prone to more iteration, large computation, slow speed and other shortcomings. For this reason, we incorporate the line search method, self-adaptive trust region method and fixed-step method with non-monotone trust region method respectively and put forward three new non-monotone trust region methods. Besides, we prove the global convergence of each algorithm. Details are as follows:Firstly, an improved non-monotone trust region method with line search strategy is proposed. The algorithm performs the non-monotone Wolfe-type line search strategy to find an iterative point instead of resolving the sub-problem if a trial step is not adopted. Thereby, the new method improves the efficiency of the algorithm effectively.Secondly, we assimilate an efficient method of self-adaption into the non-monotone trust region method in order to propose the new non-monotone adaptive trust region method. The application of non-monotone technique and adaptive updating method can reduce the number of resolving sub-problems and avoid a large amount of calculations.Thirdly, we propose and analyze a new non-monotone adaptive trust region method with fixed step-size. It utilizes a fixed formula to get the next iterative point if a trial step is not adopted and the renewal of the trust region radius takes a more computationally simple method. The new method greatly reduces the complexity of the algorithm.Finally, we summarize the proposed methods and prospect further research.