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

Jinhua Wang, Yaohua Hu (), Carisa Kwok Wai Yu and Xiaojun Zhuang
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Jinhua Wang: Hangzhou Normal University
Yaohua Hu: Shenzhen University
Carisa Kwok Wai Yu: The Hang Seng University of Hong Kong
Xiaojun Zhuang: Zhejiang University of Technology

Journal of Optimization Theory and Applications, 2019, vol. 183, issue 2, No 7, 520-534

Abstract: 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; 90C25; 90C30; 47J25 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10957-019-01563-2

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