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Newton’s Method for Solving Generalized Equations Without Lipschitz Condition

Author

Listed:
  • Jiaxi Wang

    (Yunnan Normal University)

  • Wei Ouyang

    (Yunnan Normal University
    Key Laboratory of Complex System Modeling and Application for Universities in Yunnan)

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.

Suggested Citation

  • Jiaxi Wang & Wei Ouyang, 2022. "Newton’s Method for Solving Generalized Equations Without Lipschitz Condition," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 510-532, February.
    • Handle: RePEc:spr:joptap:v:192:y:2022:i:2:d:10.1007_s10957-021-01974-0
    DOI: 10.1007/s10957-021-01974-0
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    References listed on IDEAS

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    1. Michaël Gaydu & Gilson N. Silva, 2020. "A General Iterative Procedure to Solve Generalized Equations with Differentiable Multifunction," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 207-222, April.
    2. F. Aragón Artacho & A. Belyakov & A. Dontchev & M. López, 2014. "Local convergence of quasi-Newton methods under metric regularity," Computational Optimization and Applications, Springer, vol. 58(1), pages 225-247, May.
    3. Leopoldo Marini & Benedetta Morini & Margherita Porcelli, 2018. "Quasi-Newton methods for constrained nonlinear systems: complexity analysis and applications," Computational Optimization and Applications, Springer, vol. 71(1), pages 147-170, September.
    4. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
    5. Hongjin He & Chen Ling & Hong-Kun Xu, 2015. "A Relaxed Projection Method for Split Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 213-233, July.
    Full references (including those not matched with items on IDEAS)

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