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A Modified Self-Scaling Memoryless Broyden–Fletcher–Goldfarb–Shanno Method for Unconstrained Optimization

Author

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  • C. X. Kou

    (Beijing University of Posts and Telecommunications)

  • Y. H. Dai

    (Chinese Academy of Sciences)

Abstract

The introduction of quasi-Newton and nonlinear conjugate gradient methods revolutionized the field of nonlinear optimization. The self-scaling memoryless Broyden–Fletcher–Goldfarb–Shanno (SSML-BFGS) method by Perry (Disscussion Paper 269, 1977) and Shanno (SIAM J Numer Anal, 15, 1247–1257, 1978) provided a good understanding about the relationship between the two classes of methods. Based on the SSML-BFGS method, new conjugate gradient algorithms, called CG_DESCENT and CGOPT, have been proposed by Hager and Zhang (SIAM J Optim, 16, 170–192, 2005) and Dai and Kou (SIAM J Optim, 23, 296–320, 2013), respectively. It is somewhat surprising that the two conjugate gradient methods perform more efficiently than the SSML-BFGS method. In this paper, we aim at proposing some suitable modifications of the SSML-BFGS method such that the sufficient descent condition holds. Convergence analysis of the modified method is made for convex and nonconvex functions, respectively. The numerical experiments for the testing problems from the Constrained and Unconstrained Test Environment collection demonstrate that the modified SSML-BFGS method yields a desirable improvement over CGOPT and the original SSML-BFGS method.

Suggested Citation

  • C. X. Kou & Y. H. Dai, 2015. "A Modified Self-Scaling Memoryless Broyden–Fletcher–Goldfarb–Shanno Method for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 165(1), pages 209-224, April.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:1:d:10.1007_s10957-014-0528-4
    DOI: 10.1007/s10957-014-0528-4
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    References listed on IDEAS

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    1. J. Z. Zhang & N. Y. Deng & L. H. Chen, 1999. "New Quasi-Newton Equation and Related Methods for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 147-167, July.
    2. Shmuel S. Oren & David G. Luenberger, 1974. "Self-Scaling Variable Metric (SSVM) Algorithms," Management Science, INFORMS, vol. 20(5), pages 845-862, January.
    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.
    4. Shmuel S. Oren, 1974. "Self-Scaling Variable Metric (SSVM) Algorithms," Management Science, INFORMS, vol. 20(5), pages 863-874, January.
    5. Avinoam Perry, 1977. "A Class of Conjugate Gradient Algorithms with a Two-Step Variable Metric Memory," Discussion Papers 269, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
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    Cited by:

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    3. S. Bojari & M. R. Eslahchi, 2020. "Global convergence of a family of modified BFGS methods under a modified weak-Wolfe–Powell line search for nonconvex functions," 4OR, Springer, vol. 18(2), pages 219-244, June.

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