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

1
Research Center in Mathematics and Applied Mathematics, Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
2
School of Science, University of Phayao, Phayao 56000, Thailand
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 42; https://doi.org/10.3390/math8010042
Received: 12 December 2019 / Revised: 20 December 2019 / Accepted: 21 December 2019 / Published: 1 January 2020
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. View Full-Text
Keywords: convex minimization problem; forward-backward splitting method; Hilbert space; line search convex minimization problem; forward-backward splitting method; Hilbert space; line search
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Suantai, S.; Kankam, K.; Cholamjiak, P. A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces. Mathematics 2020, 8, 42.

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