Strong convergence of the forward–backward splitting algorithms via linesearches in Hilbert spaces

Abstract

Many real-world problems in applied sciences, engineering and economics can be reformulated as the convex minimization problem of two objective functions. In order to solve this problem, the forward–backward splitting algorithm and its modifications have been invented and used to investigate the convergence analysis. In this work, we discuss strong convergence of the sequence generated by the forward–backward algorithm involving the viscosity approximation method. The stepsizes studied in this paper are defined by two different kinds of linesearches which do not require the Lipschitz continuity condition on the gradient. Finally, we provide numerical experiments to show the efficiency of the proposed methods in signal recovery. It is reported that our algorithms have a good convergence behavior in practice without information of Lipschitz constants of the gradient of functions.

Keywords:

• Forward–backward method
• strong convergence
• convex minimization
• Hilbert space

2010 AMS Subject Classifications:

• 65K05
• 90C25
• 90C30

Acknowledgements

The authors would like to thank editor and reviewers for their value comments. The authors thank School of Science, University of Phayao (the revenue budget 2021).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project is funded by National Research Council of Thailand (NRCT) [grant no. N41A640094].