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Inexact proximal methods for weakly convex functions

Author

Listed:
  • Pham Duy Khanh

    (Ho Chi Minh City University of Education)

  • Boris S. Mordukhovich

    (Wayne State University)

  • Vo Thanh Phat

    (University of North Dakota)

  • Dat Ba Tran

    (Wayne State University)

Abstract

This paper proposes and develops inexact proximal methods for finding stationary points of the sum of a smooth function and a nonsmooth weakly convex one, where an error is present in the calculation of the proximal mapping of the nonsmooth term. A general framework for finding zeros of a continuous mapping is derived from our previous paper on this subject to establish convergence properties of the inexact proximal point method when the smooth term is vanished and of the inexact proximal gradient method when the smooth term satisfies a descent condition. The inexact proximal point method achieves global convergence with constructive convergence rates when the Moreau envelope of the objective function satisfies the Kurdyka–Łojasiewicz (KL) property. Meanwhile, when the smooth term is twice continuously differentiable with a Lipschitz continuous gradient and a differentiable approximation of the objective function satisfies the KL property, the inexact proximal gradient method achieves the global convergence of iterates with constructive convergence rates.

Suggested Citation

  • Pham Duy Khanh & Boris S. Mordukhovich & Vo Thanh Phat & Dat Ba Tran, 2025. "Inexact proximal methods for weakly convex functions," Journal of Global Optimization, Springer, vol. 91(3), pages 611-646, March.
    • Handle: RePEc:spr:jglopt:v:91:y:2025:i:3:d:10.1007_s10898-024-01460-7
    DOI: 10.1007/s10898-024-01460-7
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    References listed on IDEAS

    as
    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    3. Pham Duy Khanh & Boris S. Mordukhovich & Vo Thanh Phat & Dat Ba Tran, 2023. "Generalized damped Newton algorithms in nonsmooth optimization via second-order subdifferentials," Journal of Global Optimization, Springer, vol. 86(1), pages 93-122, May.
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