A new nonmonotone line-search trust-region approach for nonlinear systems

This paper introduces a new derivative-free trust-region algorithm for solving nonlinear systems, based on a new nonmonotone technique and an adaptive radius strategy. It is shown that we can generate the small (large) steps and radii in the cases where iterations are near (far away from) the optimizer. Such a nonmonotone strategy is embedded into the trust region framework and Armijo line search to face with problems which have the narrow curved valley. To prevent resolving the trust-region subproblem, the nonmonotone Armijo line search is used whenever iterations are unsuccessful. In each iteration, the adaptive radius strategy is constructed based on the norm of the best function values. The global and q-quadratic rate of convergence of the new algorithm is proved. Computational results are reported.