A new iterative method for solving the multiple-set split variational inequality problem in Hilbert spaces

Abstract

This paper proposes a new self-adaptive algorithm for solving the multiple-set split variational inequality problem in Hilbert spaces. Our algorithm uses dynamic step-sizes, chosen based on information of the previous step. In comparison with the work by Censor et al. [Numer Algorithms. 2012;59:301–323], the new algorithm gives strong convergence results and does not require information about the transformation operator's norm. Some applications of our main results regarding the solution of the multiple-set split feasibility problem and the split feasibility problem are presented and show that the iterative method converges strongly under weaker assumptions than the ones used recently by Xu [Inverse Probl. 2006;22:2021–2034] and by Buong [Numer Algorithms. 2017;76:783–798]. Numerical experiments on finite-dimensional and infinite-dimensional spaces and an application to discrete optimal control problems are reported to demonstrate the advantages and efficiency of the proposed algorithms over some existing results.