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Rates of convergence of Newton type methods for variational inequalities and nonlinear programming

Abstract : This paper presents some new results in the theory of Newton type methods for variational inequalities and their application to nonlinear programming. A condition of semi-stability is shown to ensure the quadratic convergence of Newton's method and the superlinear convergence of some quasi-Newton algorithms, provided the sequence defined by the algorithm exists and converges. A partial extension of these results to nonsmooth function is given. The second part of the paper considers some particular variationnal inequalities with unknowns {x, l) generalizing optimality systems. Here only the question of superlinear convergence of {xk} is considered. Some necessary or sufficient conditions are given. Applied to some quasi-Newton algorithms they allow to obtain the superlinear convergence of {xk}. The application of the previous results to nonlinear programming allows to strenghten the know results, the main point being a characterization of the superlinear convergence of {xk} assuming a weak second-order condition without strict complementary.
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https://hal.inria.fr/inria-00075298
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Submitted on : Wednesday, May 24, 2006 - 5:53:57 PM
Last modification on : Friday, February 4, 2022 - 3:23:55 AM
Long-term archiving on: : Tuesday, April 12, 2011 - 10:24:12 PM

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  • HAL Id : inria-00075298, version 1

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J. Frederic Bonnans. Rates of convergence of Newton type methods for variational inequalities and nonlinear programming. [Research Report] RR-1260, INRIA. 1990. ⟨inria-00075298⟩

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