Robust Stopping Criteria for Dykstra's Algorithm

Dykstra's algorithm is a suitable alternating projection scheme for solving the optimization problem of finding the closest point to one given in the intersection of a finite number of closed and convex sets. It has been recently used in a wide variety of applications. However, in practice, the commonly used stopping criteria are not robust and could stop the iterative process prematurely at a point that does not solve the optimization problem. In this work we present a counterexample to illustrate the weakness of the commonly used criteria, and then we develop robust stopping rules. Additional experimental results are shown to illustrate the advantages of this new stopping criteria, including their associated computational cost.