Generalized equations and the generalized Newton method

The paper gives some convergence results on the generalized Newton method (referred to by some authors as Newton’s method) and the chord method when applied to generalized equations. The main results of the paper extend the classical Kantorovich results on Newton’s method to (nonsmooth) generalized equations. The present results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan. The authors also propose inner-iterative schemes for the computation of the generalized Newton iterates. These schemes generalize popular iterative methods (Richardson’s method, Jacobi’s method and the Gauss-Seidel method) for the solution of linear equations and linear complementarity problems and are shown to be convergent under natural generalizations of classical convergence criteria. The present results are applicable to equations involving single-valued functions and also to a class of generalized equations which includes variational inequalities, nonlinear complementarity problems and some nonsmooth convex minimization problems.