The convex feasibility problem consists in finding a point in the intersection of a finite family of closed convex sets. When the intersection is empty, a best compromise is to search for a point that minimizes the sum of the squared distances to the sets. In 2001, Alvaro de Pierro conjectured that the limit cycles generated by the -under-relaxed cyclic projection method converge when towards a least squares solution. While the conjecture has been confirmed under fairly general conditions, we show that the property is false in general by constructing a system of three compact convex sets in for which the -under-relaxed cycles do not converge.
A key feature of this construction is that for any chosen value of the under-relaxed projections between , and converge to the same unique -cycle, no matter what starting point is chosen. This cycle is located in a horisontal plane, with the z-coordinate depending on . As these limit cycles oscillate between the planes and , and hence contradict the conjecture, as the relevant limit does not exist.
This is joint work with Roberto Cominetti and Andrew Williamson, see arXiv:1801.03216.