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An Efficient Gradient Method with Approximately Optimal Stepsize Based on Tensor Model for Unconstrained Optimization

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  • Zexian Liu

    (Xidian University
    Hezhou University)

  • Hongwei Liu

    (Xidian University)

Abstract

A new type of stepsize, which was recently introduced by Liu et al. (Optimization 67(3):427–440, 2018), is called approximately optimal stepsize and is very efficient for gradient method. Interestingly, all gradient methods can be regarded as gradient methods with approximately optimal stepsizes. In this paper, we present an efficient gradient method with approximately optimal stepsize based on tensor model for unconstrained optimization. In the proposed method, if the objective function is not close to a minimizer and a quadratic function on a line segment between the current and latest iterates, then a tensor model is exploited to generate approximately optimal stepsize for gradient method. Otherwise, quadratic approximation models are constructed to generate approximately optimal stepsizes for gradient method. The global convergence of the proposed method is established under weak conditions. Numerical results indicate that the proposed method is very promising.

Suggested Citation

  • Zexian Liu & Hongwei Liu, 2019. "An Efficient Gradient Method with Approximately Optimal Stepsize Based on Tensor Model for Unconstrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 608-633, May.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-019-01475-1
    DOI: 10.1007/s10957-019-01475-1
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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. Hongwei Liu & Xiangli Li, 2013. "Modified subspace Barzilai-Borwein gradient method for non-negative matrix factorization," Computational Optimization and Applications, Springer, vol. 55(1), pages 173-196, May.
    3. Gonglin Yuan & Zehong Meng & Yong Li, 2016. "A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 129-152, January.
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    5. Gonglin Yuan & Zengxin Wei, 2010. "Convergence analysis of a modified BFGS method on convex minimizations," Computational Optimization and Applications, Springer, vol. 47(2), pages 237-255, October.
    6. 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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    Cited by:

    1. Zexian Liu & Hongwei Liu & Yu-Hong Dai, 2020. "An improved Dai–Kou conjugate gradient algorithm for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 75(1), pages 145-167, January.

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