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Another Conjugate Gradient Algorithm with Guaranteed Descent and Conjugacy Conditions for Large-scale Unconstrained Optimization

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  • Neculai Andrei

    (Center for Advanced Modeling and Optimization
    Academy of Romanian Scientists)

Abstract

In this paper, we suggest another accelerated conjugate gradient algorithm for which both the descent and the conjugacy conditions are guaranteed. The search direction is selected as a linear combination of the gradient and the previous direction. The coefficients in this linear combination are selected in such a way that both the descent and the conjugacy condition are satisfied at every iteration. The algorithm introduces the modified Wolfe line search, in which the parameter in the second Wolfe condition is modified at every iteration. It is shown that both for uniformly convex functions and for general nonlinear functions, the algorithm with strong Wolfe line search generates directions bounded away from infinity. The algorithm uses an acceleration scheme modifying the step length in such a manner as to improve the reduction of the function values along the iterations. Numerical comparisons with some conjugate gradient algorithms using a set of 75 unconstrained optimization problems with different dimensions show that the computational scheme outperforms the known conjugate gradient algorithms like Hestenes and Stiefel; Polak, Ribière and Polyak; Dai and Yuan or the hybrid Dai and Yuan; CG_DESCENT with Wolfe line search, as well as the quasi-Newton L-BFGS.

Suggested Citation

  • Neculai Andrei, 2013. "Another Conjugate Gradient Algorithm with Guaranteed Descent and Conjugacy Conditions for Large-scale Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 159-182, October.
    • Handle: RePEc:spr:joptap:v:159:y:2013:i:1:d:10.1007_s10957-013-0285-9
    DOI: 10.1007/s10957-013-0285-9
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    References listed on IDEAS

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    1. Avinoam Perry, 1978. "Technical Note—A Modified Conjugate Gradient Algorithm," Operations Research, INFORMS, vol. 26(6), pages 1073-1078, December.
    2. David F. Shanno, 1978. "Conjugate Gradient Methods with Inexact Searches," Mathematics of Operations Research, INFORMS, vol. 3(3), pages 244-256, August.
    3. Y.H. Dai & Y. Yuan, 2001. "An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimization," Annals of Operations Research, Springer, vol. 103(1), pages 33-47, March.
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    Cited by:

    1. Hongwei Liu & Zexian Liu, 2019. "An Efficient Barzilai–Borwein Conjugate Gradient Method for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 879-906, March.
    2. Dong, Xiao Liang & Liu, Hong Wei & He, Yu Bo, 2015. "New version of the three-term conjugate gradient method based on spectral scaling conjugacy condition that generates descent search direction," Applied Mathematics and Computation, Elsevier, vol. 269(C), pages 606-617.
    3. XiaoLiang Dong & Hongwei Liu & Yubo He, 2015. "A Self-Adjusting Conjugate Gradient Method with Sufficient Descent Condition and Conjugacy Condition," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 225-241, April.
    4. 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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