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A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces

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  • Suthep Suantai

    (Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand)

  • Kunrada Kankam

    (School of Science, University of Phayao, Phayao 56000, Thailand)

  • Prasit Cholamjiak

    (School of Science, University of Phayao, Phayao 56000, Thailand)

Abstract

In this work, we aim to investigate the convex minimization problem of the sum of two objective functions. This optimization problem includes, in particular, image reconstruction and signal recovery. We then propose a new modified forward-backward splitting method without the assumption of the Lipschitz continuity of the gradient of functions by using the line search procedures. It is shown that the sequence generated by the proposed algorithm weakly converges to minimizers of the sum of two convex functions. We also provide some applications of the proposed method to compressed sensing in the frequency domain. The numerical reports show that our method has a better convergence behavior than other methods in terms of the number of iterations and CPU time. Moreover, the numerical results of the comparative analysis are also discussed to show the optimal choice of parameters in the line search.

Suggested Citation

  • Suthep Suantai & Kunrada Kankam & Prasit Cholamjiak, 2020. "A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces," Mathematics, MDPI, vol. 8(1), pages 1-13, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:42-:d:304116
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    References listed on IDEAS

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    1. 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.
    2. Prasit Cholamjiak & Suthep Suantai, 2012. "Viscosity approximation methods for a nonexpansive semigroup in Banach spaces with gauge functions," Journal of Global Optimization, Springer, vol. 54(1), pages 185-197, September.
    3. 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.
    4. Unknown, 2005. "Forward," 2005 Conference: Slovenia in the EU - Challenges for Agriculture, Food Science and Rural Affairs, November 10-11, 2005, Moravske Toplice, Slovenia 183804, Slovenian Association of Agricultural Economists (DAES).
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    1. Adisak Hanjing & Suthep Suantai, 2023. "Novel Algorithms with Inertial Techniques for Solving Constrained Convex Minimization Problems and Applications to Image Inpainting," Mathematics, MDPI, vol. 11(8), pages 1-18, April.
    2. Adisak Hanjing & Limpapat Bussaban & Suthep Suantai, 2022. "The Modified Viscosity Approximation Method with Inertial Technique and Forward–Backward Algorithm for Convex Optimization Model," Mathematics, MDPI, vol. 10(7), pages 1-16, March.

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