There is no variational characterization of the cycles in the method of periodic projections (Q652441): Difference between revisions

The title of this paper meets two old questions, namely first, given a force, is there a potential, belonging to this force, or, given a differential equation, can it be seen as Euler's equation of a variational problem? And secondly, what about the behaviour of alternating projections on more than two nonempty closed convex (ordered) possibly nonintersecting subsets of a Hilbert space? As for the second problem remember Schwarz's alternating method, there are also (for two sets) old results of J. von Neumann, mentioned in this paper. The authors' main result is that for more than two sets in Hilbert spaces with more than two dimensions there is (under conditions) no function \(F\), such that the solution points of the cyclic projections on that sets are the unique solution of a constrained variational problem with \(F\). The proof is given by contradiction using Froda's theorem. This theorem answers a long standing question from the sixties in the negative. By the way, the authors mention that for two sets there are (under assumptions) even explicit results; there are variational problems characterizing the alternating projections. Some remarks about motivations and related projection algorithms are added.