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Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization

Author

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  • Yakui Huang

    (Xidian University
    LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

  • Hongwei Liu

    (Xidian University)

Abstract

We present a smoothing projected Barzilai–Borwein (SPBB) algorithm for solving a class of minimization problems on a closed convex set, where the objective function is nonsmooth nonconvex, perhaps even non-Lipschitz. At each iteration, the SPBB algorithm applies the projected gradient strategy that alternately uses the two Barzilai–Borwein stepsizes to the smooth approximation of the original problem. Nonmonotone scheme is adopted to ensure global convergence. Under mild conditions, we prove convergence of the SPBB algorithm to a scaled stationary point of the original problem. When the objective function is locally Lipschitz continuous, we consider a general constrained optimization problem and show that any accumulation point generated by the SPBB algorithm is a stationary point associated with the smoothing function used in the algorithm. Numerical experiments on $$\ell _2$$ ℓ 2 - $$\ell _p$$ ℓ p problems, image restoration problems, and stochastic linear complementarity problems show that the SPBB algorithm is promising.

Suggested Citation

  • Yakui Huang & Hongwei Liu, 2016. "Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization," Computational Optimization and Applications, Springer, vol. 65(3), pages 671-698, December.
    • Handle: RePEc:spr:coopap:v:65:y:2016:i:3:d:10.1007_s10589-016-9854-9
    DOI: 10.1007/s10589-016-9854-9
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    References listed on IDEAS

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    6. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. 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.
    2. Yu-Hong Dai & Yakui Huang & Xin-Wei Liu, 2019. "A family of spectral gradient methods for optimization," Computational Optimization and Applications, Springer, vol. 74(1), pages 43-65, September.
    3. Jinman Lv & Zhenhua Peng & Zhongping Wan, 2021. "Optimality Conditions, Qualifications and Approximation Method for a Class of Non-Lipschitz Mathematical Programs with Switching Constraints," Mathematics, MDPI, vol. 9(22), pages 1-20, November.

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