A New Nonmonotone Adaptive Retrospective Trust Region Method for Unconstrained Optimization Problems

In this paper, we propose a new nonmonotone adaptive retrospective Trust Region (TR) method for solving unconstrained optimization problems. Inspired by the retrospective ratio proposed in Bastin et al. (Math Program Ser A 123:395–418, 2010 ), a new nonmonotone TR ratio is introduced based on a convex combination of the nonmonotone classical and retrospective ratios. Due to the value of the new ratio, the TR radius is updated adaptively by a variant of the rule as proposed in Shi and Guo (J Comput Appl Math 213:509–520, 2008 ). Global convergence property of the new algorithm, as well as its superlinear convergence rate, is established under some standard assumptions. Numerical results on some test problems show the efficiency and effectiveness of the new method in practice, too.