Projection methods for finding the greatest element of the intersection of max-closed convex sets

We focus on the problem of finding the greatest element of the intersection of max-closed convex sets. For this purpose, we analyze the famous method of cyclic projections and show that, if this method is suitably initialized and applied to max-closed convex sets, it converges to the greatest element of their intersection. Moreover, we propose another projection method, called the decreasing projection, which turns out both theoretically and practically preferable to Euclidean projections in this particular case. Next, we argue that several known algorithms, such as Bellman-Ford and Floyd-Warshall algorithms for shortest paths or Gauss-Seidel variant of value iteration in Markov decision processes, can be interpreted as special cases of iteratively applying decreasing projections onto certain max-closed convex sets. Finally, we link decreasing projections (and thus also the aforementioned algorithms) to bounds consistency in constraint programming.