Globalizing convergence of piecewise Newton methods

Abstract

We consider versions of the Newton method for piecewise smooth nonlinear equations, as well as of the Gauss–Newton method for the case when additional constraints are imposed, supplied with linesearch procedures for the residual of the equation, aiming at globalization of convergence. (Constrained) piecewise smooth nonlinear equations arise naturally as reformulations of systems of equations and inequalities involving complementarity conditions. In cases when the direction of the Newton method cannot be computed, or appears too long, the algorithm switches to a safeguarding step of the gradient descent method for the squared residual of of the equation with smooth selection mapping active at the current iterate. For the Gauss–Newton method, safeguarding steps of the gradient projection method are employed. We obtain results characterizing properties of possible accumulation points of sequences generated by these methods, namely, stationarity of any such point for at least one smooth selection mapping active at it, and conditions assuring asymptotic superlinear convergence rate of such sequences. Special attention is paid to the majorization condition for the norm of the mapping by the norms of smooth selection mappings, playing a crucial role in the analysis for the piecewise smooth case. Examples are provided demonstrating that in cases of violation of this condition, the algorithms in question may produce sequences converging to points that are not stationary for any active smooth selection mapping.