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

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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 \(\alpha \in [0,1]\) 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 \(\frac{1}{2}\) hold at a local minimum point.

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Acknowledgements

The authors would like to thank the anonymous referees and the associate editor for their valuable suggestions, which have helped to improve the presentation of this manuscript. Bai’s work was supported in part by Chongqing University of Arts and Sciences (M2019 ME07) and Chongqing Jiaotong University (2020S0055). Li’s work was supported in part by Chongqing National Applied Mathematics Center (CQ-NCAM-2021-02) and the foundation for high-level talents of Chongqing University of Art and Sciences (P2017SC01). Zhu’s work was supported by Natural Science Foundation of China (71871140).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, China

    Sixuan Bai

  2. The Key Laboratory of Complex Data Analysis and Artificial Intelligence of Chongqing, Chongqing University of Arts and Sciences, Yongchuan, Chongqing, China

    Minghua Li & Chengwu Lu

  3. Antai College of Economics and Management and Sino-US Global Logistics Institute, Shanghai, China

    Daoli Zhu

  4. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, USA

    Sien Deng

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  1. Sixuan Bai
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  2. Minghua Li
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  3. Chengwu Lu
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  4. Daoli Zhu
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Correspondence to Minghua Li.

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Communicated by Aris Daniilidis.

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Bai, S., Li, M., Lu, C. et al. The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions. J Optim Theory Appl 194, 220–245 (2022). https://doi.org/10.1007/s10957-022-02016-z

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