Sparse string-averaging and split common fixed points

Abstract

We review the common fixed point problem for the class of directed operators. This class is important because many commonly used nonlinear operators in convex optimization belong to it. We present our recent definition of sparseness of a family of operators and discuss a string-averaging algorithmic scheme that favorably handles the common fixed points problem when the family of operators is sparse. We also review some recent results on the multiple operators split common fixed point problem which requires to find a common fixed point of a family of operators in one space whose image under a linear transformation is a common fixed point of another family of operators in the image space.

Iterative methods for solving variational inequalities in Euclidean space

“…Yamada and Ogura called the operators firmly quasi-nonexpansive [50]. Zaknoon [52], and Segal and Censor [42,19,20] called cutters directed operators. In [10] these operators were called separating operators.…”

Extrapolation and local acceleration of an iterative process for common fixed point problems

“…Yamada and Ogura [YO04] and Mainge [Mai10] named the cutters firmly quasinonexpansive operators. These operators were named directed operators in Zaknoon [Zak03] and further employed under this name by Segal [Seg08] and Censor and Segal [CS08,CS08a,CS09]. Cegielski [Ceg10, Definition 2.1] named and studied these operators as separating operators.…”

Viscosity and inertial algorithms for the split common fixed point problem with applications to compressed sensing

“…In recent years, the MSCFPP was under the spotlight of research, and many scholars proposed numerous efficient algorithms. For various convergence results, we refer to [2,3,4,5,6] and the references therein.…”