Newton’s Method for Solving Generalized Equations Without Lipschitz Condition

Jiaxi Wang () and Wei Ouyang
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Jiaxi Wang: Yunnan Normal University
Wei Ouyang: Yunnan Normal University

Journal of Optimization Theory and Applications, 2022, vol. 192, issue 2, No 5, 510-532

Abstract: 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.

Keywords: Newton’s method; generalized equation; (Strong); Hölder calmness; 49J53; 49M15; 65K10; 90C48 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-021-01974-0

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