A New Nonmonotone Trust Region Method

Trust region method is an effective numerical method for unconstrained optimization problems. Trust region method is reliable and has new idea, strongly convergence. Not only it can quickly resolve the issue of good states, but also can effectively solve the ill-conditioned problems. So it wins the good graces of many experts and scholars. In recent decade, the research on monotone trust region has matured and theory is very perfect. Since Professor Naiyang Deng and other people proposed a class of non-monotone trust region methods in 1993, although the research on nonmonotone trust region method has obtained some achievements, such as adaptive nonmonotone trust region method, quasi-Newton nonmonotone trust region, nonmonotone trust region method with fixed stepsize. Theory is not perfect, so it remains further study.Of the existing nonmonotone trust region method, commonly used reference function values have f_l(k), f_k^r and C_k. A lot of research and numerical experiments show that the algorithm which uses f_l(k) as reference function value has two major issues: First, the numerical result depends on the selection of M; Second, Reference function value is likely much larger than the function value, which is bad for convergence rate. The literatures which use f_k^r as reference function value are less, so it is worth further study. The method which uses C_k as reference function value with nonmonotone Wolfe line search has strongly convergence and better numerical experiment results.In chapter 2, on the basis of the nonmonotone line search technique proposed by Zhang and Hanger (2004), a new nonmonotone trust region method is proposed by changing the predicted reduction. The global convergence and superlinear convergence results of algorithm are novel proved under proper conditions. Finally, several numerical experiment results of typical test functions show the new algorithm is effective.In chapter 3, on the basis of the new method proposed in chapter 2, a new nonmonotone trust region method is proposed by using nonmonotone Wolfe line search to ensure { B_k } is positive definite. The global convergence of algorithm is novel proved under proper conditions. Numerical experiment results show the effectiveness of new algorithm.