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Error Bound and Isocost Imply Linear Convergence of DCA-Based Algorithms to D-Stationarity

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  • Min Tao

    (Nanjing University)

  • Jiang-Ning Li

    (Nanjing University)

Abstract

We consider a class of structured nonsmooth difference-of-convex minimization, which can be written as the difference of two convex functions possibly nonsmooth with the second one in the format of the maximum of a finite convex smooth functions. We propose two extrapolation proximal difference-of-convex-based algorithms for potential acceleration to converge to a weak/standard d-stationary point of the structured nonsmooth problem, and prove its linear convergence of these algorithms under the assumptions of piecewise error bound and piecewise isocost condition. As a product, we refine the linear convergence analysis of sDCA and $$\varepsilon $$ ε -DCA in a recent work of Dong and Tao (J Optim Theory Appl 189: 190–220, 2021) by removing the assumption of locally linear regularity regarding the intersection of certain stationary sets and dominance regions. We also discuss sufficient conditions to guarantee these assumptions and illustrate that several sparse learning models satisfy all these assumptions. Finally, we conduct some elementary numerical simulations on sparse recovery to verify the theoretical results empirically.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:1:d:10.1007_s10957-023-02171-x
    DOI: 10.1007/s10957-023-02171-x
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. Tianxiang Liu & Ting Kei Pong, 2017. "Further properties of the forward–backward envelope with applications to difference-of-convex programming," Computational Optimization and Applications, Springer, vol. 67(3), pages 489-520, July.
    7. 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.
    8. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
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