Abstract
This paper aims to establish higher order convergence of the (inexact) Newton’s method for solving generalized equations composed of the sum of a single-valued mapping and a set-valued mapping between arbitrary Banach spaces without Lipschitz conditions. Imposing Hölder calmness property on the gradient of the single-valued mapping instead of Lipschitz continuity, by virtue of the contraction mapping principle, we establish exact relationship between the order of calmness for the gradient and the order of local convergence for the (inexact) Newton’s method. Furthermore, we extend the obtained results to a restricted version of the Newton’s method, which ensures that every sequence generated by this method converges to a solution of the generalized equation. Numerical examples are provided to illustrate the theoretical results.
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Acknowledgements
The authors would like to express their grate gratitude to the editors and referees for their helpful comments and constructive suggestions which help us to improve the quality of this work. This research was supported by the National Natural Science Foundation of China (Grants 11801500).
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Communicated by Juan Parra.
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Wang, J., Ouyang, W. Newton’s Method for Solving Generalized Equations Without Lipschitz Condition. J Optim Theory Appl 192, 510–532 (2022). https://doi.org/10.1007/s10957-021-01974-0
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DOI: https://doi.org/10.1007/s10957-021-01974-0