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Contractive Difference-of-Convex Algorithms

Author

Listed:
  • Songnian He

    (Civil Aviation University of China)

  • Qiao-Li Dong

    (Civil Aviation University of China)

  • Michael Th. Rassias

    (Hellenic Military Academy
    Institute for Advanced Study, Program in Interdisciplinary Studies)

Abstract

The difference-of-convex algorithm (DCA) and its variants are the most popular methods to solve the difference-of-convex optimization problem. Each iteration of them is reduced to a convex optimization problem, which generally needs to be solved by iterative methods such as proximal gradient algorithm. However, these algorithms essentially belong to some iterative methods of fixed point problems of averaged mappings, and their convergence speed is generally slow. Furthermore, there is seldom research on the termination rule of these iterative algorithms solving the subproblem of DCA. To overcome these defects, we firstly show that the subproblem of the linearized proximal method (LPM) in each iteration is equal to the fixed point problem of a contraction. Secondly, by using Picard iteration to approximately solve the subproblem of LPM in each iteration, we propose a contractive difference-of-convex algorithm (cDCA) where an adaptive termination rule is presented. Both global subsequential convergence and global convergence of the whole sequence of cDCA are established. Finally, preliminary results from numerical experiments are promising.

Suggested Citation

  • Songnian He & Qiao-Li Dong & Michael Th. Rassias, 2025. "Contractive Difference-of-Convex Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-21, July.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02689-2
    DOI: 10.1007/s10957-025-02689-2
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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. Hong-Kun Xu, 2011. "Averaged Mappings and the Gradient-Projection Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 150(2), pages 360-378, August.
    3. 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.
    4. 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.
    5. Q. L. Dong & J. Z. Huang & X. H. Li & Y. J. Cho & Th. M. Rassias, 2019. "MiKM: multi-step inertial Krasnosel’skiǐ–Mann algorithm and its applications," Journal of Global Optimization, Springer, vol. 73(4), pages 801-824, April.
    6. Hoang Tuy, 2016. "Convex Analysis and Global Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-31484-6, October.
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