On the Convergence of Recursive Trust-Region Methods for Multiscale Nonlinear Optimization and Applications to Nonlinear Mechanics

We prove new convergence results for a class of multiscale trust–region algorithms introduced by Gratton et al. in [GST06] to solve unconstrained minimization problems within the Euclidean space R. We will state less restrictive assumptions on the function which has to be minimized and on the iteratively computed trust–region corrections, which allow for proving first– and second–order convergence and, moreover, locally quadratic convergence. We show the efficiency and robustness of our approach by means of examples from nonlinear continuum mechanics. Numerical results in 3d for Ogden materials are presented.