Alternating Projection Methods - Failure in the Absence of Convexity

Given an initial point, x0, sets Si for i = 1, . . . , r and their corresponding nearest point projections PSi , the method of alternating projections (MAP) attempts to finds a point in ∩i=1Si by cyclically projecting onto the sets. The original alternating projection result was due to von Neumann (1933) who was able to prove that if the sets are subspaces MAP converges in norm to P∩i=1Si . Using MAP a point in the intersection can be obtained when only the individual projection onto each of the sets are known. When the underlying sets are convex, MAP and its variants are fairly well understood. However, despite adequate theoretical justification, these techniques are routinely applied to problems involving one or more non-convex sets with good results. In this report we examine some of the difficulties encountered when dropping the assumption of convexity. This is achieved by developing a visual tool to investigate specific examples using interactive geometry package Cinderella [5].