Abstract
In this work, we provide a note on the spectral gradient projection method for solving nonlinear equations. Motivated by recent extensions of the spectral gradient method for solving nonlinear monotone equations with convex constraints, in this paper, we note that choosing the search direction as a convex combination of two different positive spectral coefficients multiplied with the residual vector is more efficient and robust compared with the standard choice of spectral gradient coefficients combined with the projection strategy of Solodov and Svaiter (A globally convergent inexact newton method for systems of monotone equations. In: Reformulation: Nonsmooth. Piecewise Smooth, Semismooth and Smoothing Methods, pp 355–369. Springer, 1998). Under suitable assumptions, the convergence of the proposed method is established. Preliminary numerical experiments show that the method is promising. In this paper, the proposed method was used to recover sparse signal and restore blurred image arising from compressive sensing.
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Acknowledgements
We thank the anonymous referees for helpful suggestions that improved the paper. We are indebted to Associate Professor Mark Dougherty affiliated at the Auburn University, USA, for his proofreading and helpful comments. We would like to thank Professor Jinkui Liu affiliated at the Chongqing Three Gorges University, Chongqing, China, for providing us access to the MATLAB source codes for SGCS and PCG methods. The first author was supported by the Petchra Pra Jom Klao Doctoral Scholarship Academic for Ph.D. Program at KMUTT (2017–2020).
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Communicated by Natasa Krejic.
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Abubakar, A.B., Kumam, P. & Mohammad, H. A note on the spectral gradient projection method for nonlinear monotone equations with applications. Comp. Appl. Math. 39, 129 (2020). https://doi.org/10.1007/s40314-020-01151-5
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DOI: https://doi.org/10.1007/s40314-020-01151-5
Keywords
- Nonlinear monotone equations
- Spectral gradient method
- Projection method
- Global convergence
- Compressive sensing