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 originally introduced by Gratton, Sartenaer, and Toint in [SIAM J. Optim., 19 (2008), pp. 414–444] to solve unconstrained minimization problems within the Euclidean space Rⁿ. We will state less restrictive assumptions on the objective function and on the iteratively computed trust-region corrections, which allow for proving first-order convergence. Moreover, we propose a novel projection approach for obtaining the initial coarse level iterates, which are needed within the nonlinear multiscale iteration. We show the efficiency and robustness of our approach by means of numerical examples from nonlinear continuum mechanics, where stored energy functions for materials of Ogden-type and for materials with visco-plastic behavior are minimized.