Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

Abstract: The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent CFPs exists and is mainly focused on two directions. One is oriented toward defining other solution concepts that will apply, such as proximity function minimization wherein a proximity function measures in some way the total violation of all constraints. The second direction investigates the behavior of algorithms that are designed to solve a consistent CFP when applied to inconsistent problems. This direction is fueled by situations wherein one lacks a priori information about the consistency or inconsistency of the CFP or does not wish to invest computational resources to get hold of such knowledge prior to running his algorithm. In this paper we bring under one roof and telegraphically review some recent works on inconsistent CFPs.