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A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations

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  • Dai, Zhifeng
  • Chen, Xiaohong
  • Wen, Fenghua

Abstract

In this paper, we propose a derivative-free method for solving large-scale nonlinear monotone equations. It combines the modified Perry’s conjugate gradient method (I.E. Livieris, P. Pintelas, Globally convergent modified Perrys conjugate gradient method, Appl. Math. Comput., 218 (2012) 9197–9207) for unconstrained optimization problems and the hyperplane projection method (M.V. Solodov, B.F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in: M. Fukushima, L. Qi (Eds.), Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Kluwer Academic Publishers, 1998, pp. 355–369). We prove that the proposed method converges globally if the equations are monotone and Lipschitz continuous without differentiability requirement on the equations, which makes it possible to solve some nonsmooth equations. Another good property of the proposed method is that it is suitable to solve large-scale nonlinear monotone equations due to its lower storage requirement. Preliminary numerical results show that the proposed method is promising.

Suggested Citation

  • Dai, Zhifeng & Chen, Xiaohong & Wen, Fenghua, 2015. "A modified Perry’s conjugate gradient method-based derivative-free method for solving large-scale nonlinear monotone equations," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 378-386.
    • Handle: RePEc:eee:apmaco:v:270:y:2015:i:c:p:378-386
    DOI: 10.1016/j.amc.2015.08.014
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    References listed on IDEAS

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    1. Jian Liu & Mengxian Tao & Chaoqun Ma & Fenghua Wen, 2014. "Utility indifference pricing of convertible bonds," International Journal of Information Technology & Decision Making (IJITDM), World Scientific Publishing Co. Pte. Ltd., vol. 13(02), pages 429-444.
    2. Avinoam Perry, 1978. "Technical Note—A Modified Conjugate Gradient Algorithm," Operations Research, INFORMS, vol. 26(6), pages 1073-1078, December.
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    Cited by:

    1. XiaoLiang Dong & Deren Han & Zhifeng Dai & Lixiang Li & Jianguang Zhu, 2018. "An Accelerated Three-Term Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 944-961, December.
    2. Dai, Zhifeng & Zhou, Huiting & Wen, Fenghua & He, Shaoyi, 2020. "Efficient predictability of stock return volatility: The role of stock market implied volatility," The North American Journal of Economics and Finance, Elsevier, vol. 52(C).
    3. Dai, Zhifeng & Wen, Fenghua, 2018. "Some improved sparse and stable portfolio optimization problems," Finance Research Letters, Elsevier, vol. 27(C), pages 46-52.
    4. Gao, Peiting & He, Chuanjiang & Liu, Yang, 2019. "An adaptive family of projection methods for constrained monotone nonlinear equations with applications," Applied Mathematics and Computation, Elsevier, vol. 359(C), pages 1-16.
    5. Waziri, Mohammed Yusuf & Ahmed, Kabiru & Sabi’u, Jamilu, 2019. "A family of Hager–Zhang conjugate gradient methods for system of monotone nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 645-660.
    6. Zhifeng Dai, 2017. "Comments on Hybrid Conjugate Gradient Algorithm for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 286-291, October.

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