[Paper Review] On cluster points of alternating projections

Heinz H. Bauschke, Dominikus Noll|arXiv (Cornell University)|Jul 10, 2013

Optimization and Variational Analysis4 references3 citations

TL;DR

This paper constructs explicit examples in the Euclidean plane where the method of alternating projections on nonconvex, closed sets produces sequences with a nondegenerate compact continuum as their set of cluster points. It proves that such behavior is possible when the sets are countably infinite unions of closed convex sets, but impossible when they are finite unions, resolving a long-standing question about the nature of cluster points in nonconvex feasibility problems.

ABSTRACT

Suppose that $A$ and $B$ are closed subsets of a Euclidean space such that $A\cap B eq\varnothing$, and we aim to find a point in this intersection with the help of the sequences $(a_n)_ nn$ and $(b_n)_ nn$ generated by the \emph{method of alternating projections}. It is well known that if $A$ and $B$ are convex, then $(a_n)_ nn$ and $(b_n)_ nn$ converge to some point in $A\cap B$. The situation in the nonconvex case is much more delicate. In 1990, Combettes and Trussell presented a dichotomy result that guarantees either convergence to a point in the intersection or a nondegenerate compact continuum as the set of cluster points. In this note, we construct two sets in the Euclidean plane illustrating the continuum case. The sets $A$ and $B$ can be chosen as countably infinite unions of closed convex sets. In contrast, we also show that such behaviour is impossible for finite unions.

Motivation & Objective

• To construct explicit examples of closed, nonconvex sets in R² where alternating projections yield a nondegenerate compact continuum as the set of cluster points.
• To demonstrate that such continuum behavior is possible when the sets are countably infinite unions of closed convex sets.
• To prove that the continuum case cannot occur when the sets are finite unions of nonempty closed convex sets.
• To clarify the dichotomy in convergence behavior of alternating projections in nonconvex settings, as previously suggested by Combettes and Trussell.

Proposed method

• Define a spiral curve in R² using a strictly decreasing radial function ρ(α) = 1 + exp(−α), parameterized by angle α.
• Introduce a distance threshold ε(α) = (ρ(α) − ρ(α + 2π))/2 to control proximity between points on the spiral.
• Construct a sequence of points (x_n) along the spiral such that consecutive points are separated by exactly ε(α_n), ensuring controlled convergence behavior.
• Define sets A and B as unions of alternating points from the sequence and the unit circle S, ensuring closedness and nonconvex structure.
• Use the nonexpansiveness of projections and compactness arguments to show that the sequences (a_n) and (b_n) generated by alternating projections converge to points on S.
• Apply a pigeonhole principle and distance-based contradiction to prove that finite unions of convex sets cannot yield nondegenerate continuum cluster points.

Experimental results

Research questions

• RQ1 Can explicit examples be constructed where alternating projections on nonconvex sets produce a nondegenerate compact continuum as the set of cluster points?
• RQ2 Is the continuum case in alternating projections possible when the sets are countably infinite unions of closed convex sets?
• RQ3 Can the continuum case be ruled out when the sets are finite unions of nonempty closed convex sets?
• RQ4 What conditions on the structure of the sets A and B determine whether the cluster points are a single point or a continuum?

Key findings

• The paper constructs two closed, nonconvex sets A and B in R², each a countably infinite union of closed convex sets, such that alternating projections yield sequences whose cluster points form a nondegenerate compact continuum (the unit circle S).
• For the constructed example, the sequences (a_n) and (b_n) converge to the unit circle S, with S being the exact set of cluster points.
• The construction shows that the continuum case is possible in the nonconvex setting when the sets are countably infinite unions of convex sets.
• The paper proves that if A and B are finite unions of nonempty closed convex sets, then the alternating projections sequences must converge to a single point in A ∩ B, ruling out continuum cluster points.
• The proof relies on nonexpansiveness of projections and a contradiction argument based on positive distance from limit points to sets not containing them.
• The result confirms that the dichotomy proposed by Combettes and Trussell—either convergence to a point or a nondegenerate continuum—can be realized in the continuum case with countable unions, but not with finite unions.