# A Family of Projection Gradient Methods for Solving the Multiple-Sets Split Feasibility Problem

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## Abstract

In the present paper, we explore a family of projection gradient methods for solving the multiple-sets split feasibility problem, which include the cyclic/simultaneous iteration methods introduced in Wen et al. (J Optim Theory Appl 166:844–860, 2015) as special cases. For the general case, where the involved sets are given by level sets of convex functions, the calculation of the projection onto the level sets is complicated in general, and thus, the resulting projection gradient method cannot be implemented easily. To avoid this difficulty, we introduce a family of relaxed projection gradient methods, in which the projections onto the approximated halfspaces are adopted in place of the ones onto the level sets. They cover the relaxed cyclic/simultaneous iteration methods introduced in Wen et al. (J Optim Theory Appl 166:844–860, 2015) as special cases. Global weak convergence theorems are established for these methods. In particular, as direct applications of the established theorems, our results fill some gaps and deal with the imperfections that appeared in Wen et al. (J Optim Theory Appl 166:844–860, 2015) and hence improve and extend the corresponding results therein.

## Keywords

Multiple-sets split feasibility problem Projection gradient methods Global weak convergence## Mathematics Subject Classification

90C25 90C30 47J25## Notes

### Acknowledgements

The authors are grateful to the anonymous referees and the editor for their valuable comments and suggestions that helped to improve the quality of this paper. Jinhua Wang was supported in part by the National Natural Science Foundation of China (Grant 11771397) and Zhejiang Provincial Natural Science Foundation of China (Grant LY17A010021). Yaohua Hu was supported in part by National Natural Science Foundation of China (11601343, 11601344, 11871347), Natural Science Foundation of Guangdong (2016A030310038), Natural Science Foundation of Shenzhen (JCYJ20170817100950436, JCYJ20170818091621856) and Interdisciplinary Innovation Team of Shenzhen University. Carisa Kwok Wai Yu is supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (UGC/FDS14/P02/15 and UGC/FDS14/P02/17).

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