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The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions

Author

Listed:
  • Sixuan Bai

    (Chongqing Jiaotong University)

  • Minghua Li

    (Chongqing University of Arts and Sciences)

  • Chengwu Lu

    (Chongqing University of Arts and Sciences)

  • Daoli Zhu

    (Antai College of Economics and Management and Sino-US Global Logistics Institute)

  • Sien Deng

    (Northern Illinois University)

Abstract

We start by establishing the equivalence of three types of error bounds: weak sharp minima, level-set subdifferential error bounds and Łojasiewicz (for short Ł) inequalities for weakly convex functions with exponent $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] and approximately convex functions. Then we apply these equivalence results to a class of nonconvex optimization problems, whose objective functions are the sum of a convex function and a composite function with a locally Lipschitz function and a smooth vector-valued function. Finally, applying a characterization for lower-order regularization problems, we show that the level-set subdifferential error bound with exponent 1 and the Ł inequality with exponent $$\frac{1}{2}$$ 1 2 hold at a local minimum point.

Suggested Citation

  • Sixuan Bai & Minghua Li & Chengwu Lu & Daoli Zhu & Sien Deng, 2022. "The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 220-245, July.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:1:d:10.1007_s10957-022-02016-z
    DOI: 10.1007/s10957-022-02016-z
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    References listed on IDEAS

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    1. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2014. "Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 107-132, July.
    2. Yaohua Hu & Chong Li & Kaiwen Meng & Xiaoqi Yang, 2021. "Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 853-883, April.
    3. Daoli Zhu & Sien Deng & Minghua Li & Lei Zhao, 2021. "Level-Set Subdifferential Error Bounds and Linear Convergence of Bregman Proximal Gradient Method," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 889-918, June.
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