Nonsmooth Equations: Motivation and Algorithms

This paper reports on some recent developments in the area of solving of nonsmooth equations by generalized Newton methods. The emphasis is on three topics: motivation, characterization of superlinear convergence, and a new Gauss–Newton method for solving a certain class of nonsmooth equations. The characterization of superlinear convergence extends the classical result of Dennis and Moré for smooth equations and that of Ip and Kyparisis for B-differentiable equations. The Gauss–Newton method is different from that proposed recently by Han, Pang, and Rangaraj; it uses convex quadratic programs to generate descent directions for the least-squares merit function.