On the string averaging method for sparse common fixed-point problems

We study 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 propose a definition of sparseness of a family of operators and investigate a string-averaging algorithmic scheme that favorably handles the common fixed-point problem when the family of operators is sparse. The convex feasibility problem is treated as a special case and a new subgradient projections algorithmic scheme is obtained.