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Generalized equations and the generalized Newton method

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Abstract

We give 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. Our results also extend earlier results on nonsmooth equations due to Eaves, Robinson, Josephy, Pang and Chan.

We 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.

Our 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.

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  1. Livinus U. Uko

    Present address: Mathematics Department, University of Ibadan, Ibadan, Nigeria

Authors and Affiliations

  1. International Centre For Theoretical Physics, Trieste, Italy

    Livinus U. Uko

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Uko, L.U. Generalized equations and the generalized Newton method. Mathematical Programming 73, 251–268 (1996). https://doi.org/10.1007/BF02592214

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  • DOI: https://doi.org/10.1007/BF02592214

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