A Trust-region Method using Extended Nonmonotone Technique for Unconstrained Optimization

In this paper, we present a nonmonotone trust-region algorithm for unconstrained optimization. We first introduce a variant of the nonmonotone strategy proposed by Ahookhosh and Amini cite{AhA 01} and incorporate it into the trust-region framework to construct a more efficient approach. Our new nonmonotone strategy combines the current function value with the maximum function values in some prior successful iterates. For iterates far away from the optimizer, we give a very strong nonmonotone strategy. In the vicinity of the optimizer, we have a weaker nonmonotone strategy. It leads to a medium nonmonotone strategy when iterates are not far away from or close to the optimizer. Theoretical analysis indicates that the new approach converges globally to a first-order critical point under classical assumptions. In addition, the local convergence is also studied. Extensive numerical experiments for unconstrained optimization problems are reported.