The use of additional diminishing disturbances in Fejer models of iterative algorithms

The behavior of Fejer processes with diminishing disturbances generated by a small shift in the argument of the Fejer operator is studied. It is shown that, if the operator is locally strongly Fejer, a diminishing disturbance does not prevent convergence to an attracting set. At the same time, such a disturbance can be used to furnish the process with additional properties that ensure convergence to certain subsets of the attracting set. In particular, based on this scheme, new parallel decomposition methods for optimization problems can be suggested that do not require that the constraints possess a specific structure.