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On the Nonmonotone Line Search

Author

Listed:
  • Y. H. Dai

    (Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and System Sciences, Chinese Academy of Sciences)

Abstract

The technique of nonmonotone line search has received many successful applications and extensions in nonlinear optimization. This paper provides some basic analyses of the nonmonotone line search. Specifically, we analyze the nonmonotone line search methods for general nonconvex functions along different lines. The analyses are helpful in establishing the global convergence of a nonmonotone line search method under weaker conditions on the search direction. We explore also the relations between nonmonotone line search and R-linear convergence assuming that the objective function is uniformly convex. In addition, by taking the inexact Newton method as an example, we observe a numerical drawback of the original nonmonotone line search and suggest a standard Armijo line search when the nonmonotone line search condition is not satisfied by the prior trial steplength. The numerical results show the usefulness of such suggestion for the inexact Newton method.

Suggested Citation

  • Y. H. Dai, 2002. "On the Nonmonotone Line Search," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 315-330, February.
    • Handle: RePEc:spr:joptap:v:112:y:2002:i:2:d:10.1023_a:1013653923062
    DOI: 10.1023/A:1013653923062
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    Citations

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    Cited by:

    1. M. Fatemi & N. Mahdavi-Amiri, 2012. "A filter trust-region algorithm for unconstrained optimization with strong global convergence properties," Computational Optimization and Applications, Springer, vol. 52(1), pages 239-266, May.
    2. M. Reza Peyghami & D. Ataee Tarzanagh, 2015. "A relaxed nonmonotone adaptive trust region method for solving unconstrained optimization problems," Computational Optimization and Applications, Springer, vol. 61(2), pages 321-341, June.
    3. Jiao Li & Yu-Fei Yang & Bo Yu, 2012. "A nonmonotone PSB algorithm for solving unconstrained optimization," Computational Optimization and Applications, Springer, vol. 52(1), pages 267-280, May.
    4. Ubaldo M. García Palomares, 2023. "Convergence of derivative-free nonmonotone Direct Search Methods for unconstrained and box-constrained mixed-integer optimization," Computational Optimization and Applications, Springer, vol. 85(3), pages 821-856, July.
    5. Lei Yang, 2024. "Proximal Gradient Method with Extrapolation and Line Search for a Class of Non-convex and Non-smooth Problems," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 68-103, January.
    6. Shi, Zhenjun & Wang, Shengquan, 2011. "Nonmonotone adaptive trust region method," European Journal of Operational Research, Elsevier, vol. 208(1), pages 28-36, January.
    7. Giulia Ferrandi & Michiel E. Hochstenbach & Nataša Krejić, 2023. "A harmonic framework for stepsize selection in gradient methods," Computational Optimization and Applications, Springer, vol. 85(1), pages 75-106, May.
    8. Juliano B. Francisco & Douglas S. Gonçalves & Fermín S. V. Bazán & Lila L. T. Paredes, 2020. "Non-monotone inexact restoration method for nonlinear programming," Computational Optimization and Applications, Springer, vol. 76(3), pages 867-888, July.
    9. Xiaopeng Zhao & Jen-Chih Yao, 2022. "Linear convergence of a nonmonotone projected gradient method for multiobjective optimization," Journal of Global Optimization, Springer, vol. 82(3), pages 577-594, March.
    10. O. P. Ferreira & M. Lemes & L. F. Prudente, 2022. "On the inexact scaled gradient projection method," Computational Optimization and Applications, Springer, vol. 81(1), pages 91-125, January.
    11. Qu, Shaojian & Ji, Ying & Jiang, Jianlin & Zhang, Qingpu, 2017. "Nonmonotone gradient methods for vector optimization with a portfolio optimization application," European Journal of Operational Research, Elsevier, vol. 263(2), pages 356-366.
    12. Crisci, Serena & Ruggiero, Valeria & Zanni, Luca, 2019. "Steplength selection in gradient projection methods for box-constrained quadratic programs," Applied Mathematics and Computation, Elsevier, vol. 356(C), pages 312-327.
    13. Livieris, Ioannis E. & Pintelas, Panagiotis, 2015. "A new class of nonmonotone conjugate gradient training algorithms," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 404-413.
    14. di Serafino, Daniela & Ruggiero, Valeria & Toraldo, Gerardo & Zanni, Luca, 2018. "On the steplength selection in gradient methods for unconstrained optimization," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 176-195.
    15. Marko Miladinović & Predrag Stanimirović & Sladjana Miljković, 2011. "Scalar Correction Method for Solving Large Scale Unconstrained Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 304-320, November.

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