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On the Linear Convergence to Weak/Standard d-Stationary Points of DCA-Based Algorithms for Structured Nonsmooth DC Programming

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  • Hongbo Dong

    (Sunnyvale)

  • Min Tao

    (Nanjing University)

Abstract

We consider a class of structured nonsmooth difference-of-convex minimization. We allow nonsmoothness in both the convex and concave components in the objective function, with a finite max structure in the concave part. Our focus is on algorithms that compute a (weak or standard) d(irectional)-stationary point as advocated in a recent work of Pang et al. (Math Oper Res 42:95–118, 2017). Our linear convergence results are based on direct generalizations of the assumptions of error bounds and separation of isocost surfaces proposed in the seminal work of Luo and Tseng (Ann Oper Res 46–47:157–178, 1993), as well as one additional assumption of locally linear regularity regarding the intersection of certain stationary sets and dominance regions. An interesting by-product is to present a sharper characterization of the limit set of the basic algorithm proposed by Pang et al., which fits between d-stationarity and global optimality. We also discuss sufficient conditions under which these assumptions hold. Finally, we provide several realistic and nontrivial statistical learning models where all assumptions hold.

Suggested Citation

  • Hongbo Dong & Min Tao, 2021. "On the Linear Convergence to Weak/Standard d-Stationary Points of DCA-Based Algorithms for Structured Nonsmooth DC Programming," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 190-220, April.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:1:d:10.1007_s10957-021-01827-w
    DOI: 10.1007/s10957-021-01827-w
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    References listed on IDEAS

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    1. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    2. Tianxiang Liu & Ting Kei Pong & Akiko Takeda, 2019. "A refined convergence analysis of $$\hbox {pDCA}_{e}$$ pDCA e with applications to simultaneous sparse recovery and outlier detection," Computational Optimization and Applications, Springer, vol. 73(1), pages 69-100, May.
    3. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    4. Hoai An Le Thi & Van Ngai Huynh & Tao Pham Dinh, 2018. "Convergence Analysis of Difference-of-Convex Algorithm with Subanalytic Data," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 103-126, October.
    5. Jong-Shi Pang & Meisam Razaviyayn & Alberth Alvarado, 2017. "Computing B-Stationary Points of Nonsmooth DC Programs," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 95-118, January.
    6. Bo Wen & Xiaojun Chen & Ting Kei Pong, 2018. "A proximal difference-of-convex algorithm with extrapolation," Computational Optimization and Applications, Springer, vol. 69(2), pages 297-324, March.
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    Cited by:

    1. Min Tao & Jiang-Ning Li, 2023. "Error Bound and Isocost Imply Linear Convergence of DCA-Based Algorithms to D-Stationarity," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 205-232, April.

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