Backward Step Control for Global Newton-Type Methods

We present and analyze a new damping approach called backward step control for the globalization of the convergence of Newton-type methods for the numerical solution of nonlinear root-finding problems. We provide and discuss reasonable assumptions that imply convergence of backward step control on the basis of generalized Newton paths in conjunction with a backward analysis argument. In particular, convergence to a specific solution and a-priori estimates on the residual reduction can be shown. Furthermore, we can guarantee a transition to full steps in the vicinity of a solution, which implies fast local convergence. We present two algorithmic realizations of backward step control and apply the method to more than one hundred examples, including comparisons with different globalization approaches for the minimization of the Rosenbrock function and to large-scale unconstrained optimization problems from the CUTEst benchmark library using backward step control for an inexact Newton method based on MINRES.