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Newton’s method with feasible inexact projections for solving constrained generalized equations

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Abstract

This paper aims to address a new version of Newton’s method for solving constrained generalized equations. This method can be seen as a combination of the classical Newton’s method applied to generalized equations with a procedure to obtain a feasible inexact projection. Using the contraction mapping principle, we establish a local analysis of the proposed method under appropriate assumptions, namely metric regularity or strong metric regularity and Lipschitz continuity. Metric regularity is assumed to guarantee that the method generates a sequence that converges to a solution. Under strong metric regularity, we show the uniqueness of the solution in a suitable neighborhood, and that all sequences starting in this neighborhood converge to this solution. We also require the assumption of Lipschitz continuity to establish a linear or superlinear convergence rate for the method.

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Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments, which have helped to substantially improve the presentation of the paper. In particular, we thank one of the referees for drawing our attention to the use of general procedures to find a feasible inexact projection, rather than a specific one as we were using in our first version. The authors were supported in part by CNPq Grants 305158/2014-7 and 302473/2017-3, FAPEG/GO and CAPES.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, GO, CEP 74001-970, Brazil

    Fabiana R. de Oliveira & Orizon P. Ferreira

  2. Centro das Ciências Exatas e das Tecnologias, Universidade Federal do Oeste da Bahia, Barreiras, BA, CEP 47808-021, Brazil

    Gilson N. Silva

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  1. Fabiana R. de Oliveira
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  2. Orizon P. Ferreira
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  3. Gilson N. Silva
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Correspondence to Fabiana R. de Oliveira.

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de Oliveira, F.R., Ferreira, O.P. & Silva, G.N. Newton’s method with feasible inexact projections for solving constrained generalized equations. Comput Optim Appl 72, 159–177 (2019). https://doi.org/10.1007/s10589-018-0040-0

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