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Nonlinear Stepsize Control Algorithms: Complexity Bounds for First- and Second-Order Optimality

Author

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  • Geovani Nunes Grapiglia

    (Universidade Federal do Paraná, Centro Politécnico)

  • Jinyun Yuan

    (Universidade Federal do Paraná, Centro Politécnico)

  • Ya-xiang Yuan

    (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

Abstract

A nonlinear stepsize control (NSC) framework has been proposed by Toint (Optim Methods Softw 28:82–95, 2013) for unconstrained optimization, generalizing several trust-region and regularization algorithms. More recently, worst-case complexity bounds to achieve approximate first-order optimality were proved by Grapiglia, Yuan and Yuan (Math Program 152:491–520, 2015) for the generic NSC framework. In this paper, improved complexity bounds for first-order optimality are obtained. Furthermore, complexity bounds for second-order optimality are also provided.

Suggested Citation

  • Geovani Nunes Grapiglia & Jinyun Yuan & Ya-xiang Yuan, 2016. "Nonlinear Stepsize Control Algorithms: Complexity Bounds for First- and Second-Order Optimality," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 980-997, December.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:3:d:10.1007_s10957-016-1007-x
    DOI: 10.1007/s10957-016-1007-x
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    References listed on IDEAS

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    1. Ju-liang Zhang & Yong Wang, 2003. "A new trust region method for nonlinear equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 58(2), pages 283-298, November.
    2. NESTEROV, Yurii, 2007. "Gauss-Newton scheme with worst case guarantees for global performance," LIDAM Reprints CORE 1952, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Geovani N. Grapiglia & Ekkehard W. Sachs, 2017. "On the worst-case evaluation complexity of non-monotone line search algorithms," Computational Optimization and Applications, Springer, vol. 68(3), pages 555-577, December.
    2. Geovani N. Grapiglia & Gabriel F. D. Stella, 2022. "An adaptive trust-region method without function evaluations," Computational Optimization and Applications, Springer, vol. 82(1), pages 31-60, May.

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