Generalized equations and the generalized Newton method |
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Authors: | Livinus U Uko |
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Institution: | (1) International Centre For Theoretical Physics, Trieste, Italy;(2) Present address: Mathematics Department, University of Ibadan, Ibadan, Nigeria |
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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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Keywords: | Generalized equations Generalized Newton method Variational inequalities Nonlinear complementarity problem Kantorovich theorem |
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