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On the Linear Convergence of Forward–Backward Splitting Method: Part I—Convergence Analysis

Author

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  • Yunier Bello-Cruz

    (Northern Illinois University)

  • Guoyin Li

    (University of New South Wales)

  • Tran T. A. Nghia

    (Oakland University)

Abstract

In this paper, we study the complexity of the forward–backward splitting method with Beck–Teboulle’s line search for solving convex optimization problems, where the objective function can be split into the sum of a differentiable function and a nonsmooth function. We show that the method converges weakly to an optimal solution in Hilbert spaces, under mild standing assumptions without the global Lipschitz continuity of the gradient of the differentiable function involved. Our standing assumptions is weaker than the corresponding conditions in the paper of Salzo (SIAM J Optim 27:2153–2181, 2017). The conventional complexity of sublinear convergence for the functional value is also obtained under the local Lipschitz continuity of the gradient of the differentiable function. Our main results are about the linear convergence of this method (in the quotient type), in terms of both the function value sequence and the iterative sequence, under only the quadratic growth condition. Our proof technique is direct from the quadratic growth conditions and some properties of the forward–backward splitting method without using error bounds or Kurdya-Łojasiewicz inequality as in other publications in this direction.

Suggested Citation

  • Yunier Bello-Cruz & Guoyin Li & Tran T. A. Nghia, 2021. "On the Linear Convergence of Forward–Backward Splitting Method: Part I—Convergence Analysis," Journal of Optimization Theory and Applications, Springer, vol. 188(2), pages 378-401, February.
  • Handle: RePEc:spr:joptap:v:188:y:2021:i:2:d:10.1007_s10957-020-01787-7
    DOI: 10.1007/s10957-020-01787-7
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    References listed on IDEAS

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    1. Dmitriy Drusvyatskiy & Adrian S. Lewis, 2018. "Error Bounds, Quadratic Growth, and Linear Convergence of Proximal Methods," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 919-948, August.
    2. Ion Necoara & Yurii Nesterov & François Glineur, 2019. "Linear convergence of first order methods for non-strongly convex optimization," LIDAM Reprints CORE 3000, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    5. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    6. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
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    Cited by:

    1. Yunier Bello-Cruz & Guoyin Li & Tran Thai An Nghia, 2022. "Quadratic Growth Conditions and Uniqueness of Optimal Solution to Lasso," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 167-190, July.
    2. Nguyen Huy Chieu & Nguyen Thi Quynh Trang & Ha Anh Tuan, 2022. "Quadratic Growth and Strong Metric Subregularity of the Subdifferential for a Class of Non-prox-regular Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 1081-1106, September.

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