Ball-relaxed projection algorithms for multiple-sets split feasibility problem

ABSTRACT

The multiple-sets split feasibility problem (MSSFP) requires finding a point closet to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. Motivated by the ball-relaxed projection algorithm proposed by Yu et al. for the split feasibility problem (SFP), in this paper, we introduce ball-relaxed projection algorithms for solving the MSSFP. Instead of the level sets or half-spaces, our algorithms require computing the orthogonal projections onto closed balls. We establish weak and strong convergence of the proposed algorithms to a solution of the MSSFP. Finally, we provide preliminary numerical experiments to show the efficiency and the implementation of our method.

Keywords:

• Multiple-sets split feasibility problem
• strongly convex function
• Hilbert space
• relaxed CQ algorithm
• ball
• self-adaptive technique

Acknowledgments

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. The first author was supported by the ‘Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi’ (Grant No.37/2561). Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005. The authors would like to thank editor in chief and the anonymous referees for their valuable comments and suggestions which helped to improve the original version of this paper. The work presented here was carried out in collaboration between all authors. All authors have contributed to, checked, read and approved the manuscript.

Disclosure statement

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

Additional information

Funding

The authors acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. The first author was supported by the ‘Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi’ [grant number 37/2561]. Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005.