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The forward–backward splitting method and its convergence rate for the minimization of the sum of two functions in Banach spaces

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Abstract

In this paper, the forward–backward splitting method for the minimization of the sum of two functions have been extended to Banach spaces. The regularization term of the forward–backward splitting method is constructed by the norm with p-power. We prove the functional values of the sequences generated by this method converge with an asymptotic rate \(n^{1-p}\) to the optimal value of the problem and establish the linear convergence under an error bound assumption.

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Acknowledgements

The work was supported by PHD research startup foundation of Harbin Normal University (No. XKB201804) and the National Natural Sciences Grant (No. 11871182).

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  1. School of Mathematical Sciences, Harbin Normal University, Harbin, 150025, People’s Republic of China

    Wei-Bo Guan & Wen Song

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  1. Wei-Bo Guan
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  2. Wen Song
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Correspondence to Wei-Bo Guan.

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Guan, WB., Song, W. The forward–backward splitting method and its convergence rate for the minimization of the sum of two functions in Banach spaces. Optim Lett 15, 1735–1758 (2021). https://doi.org/10.1007/s11590-020-01544-9

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  • DOI: https://doi.org/10.1007/s11590-020-01544-9

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