on determining radius in nonmonotone trust-region approaches

This paper proposes two effective nonmonotone trust-region frameworks for solving nonlinear unconstrained optimization problems while provide a new effective policy to update the trust-region radius. conventional nonmonotone trust-region algorithms apply a specific nonmonotone ratio to accept  new trial step and update the trust-region radius. this paper recommends using the nonmonotone ratio only as an acceptance criterion for a new trial step. in contrast, the monotone ratio or a hybrid of monotone and nonmonotone ratios is proposed as a criterion for updating the trust-region radius. we investigate the global convergence to first- and second-order stationary points for the proposed approaches under certain classical assumptions.  initial numerical results indicate that the proposed methods significantly enhance the performance of nonmonotone trust-region methods.