The Expected–Projection Method: Its Behavior and Applications to Linear Operator Equations and Convex Optimization

Abstract

It was shown by Butnariu and Flåm [J. Nnmer. Funct. Anal. Optim. 15: 601–636, 1995] that, under some conditions, sequences generated by the expected projection method (EPM) in Hilbert spaces approximate almost common points of measurable families of closed convex subsets provided that such points exist. In this work we study the behavior of the EPM in the more general situation when the involved sets may or may not have almost common points and we give necessary and sufficient conditions for weak and strong convergence. Also, we show how the EPM can be applied to finding solutions of linear operator equations and to solving convex optimization problems.