Cases of ineffectiveness of geometric cuts in sequential projection methods

Abstract

We show cases where geometric cuts, even cumulated, can be ineffective in accelerating sequential projection methods for feasibility problems in finite-dimensional spaces. Such cases are well-known in infinite- (or very-many-) dimensional spaces. There we can construct counterexamples in which the convergence of a sequential projection algorithm is very slow, merely laying at the bound induced by the Fejér contraction property, and introducing geometric cuts does not help. These counterexamples are very simple: the algorithm starts at the zero point, each step consist in moving by a new axis versor. In this paper, we construct a corresponding counterexample in a (essentially) finite-dimensional space, showing that the the possibility of ineffectiveness of geometric cuts is not connected with the space dimensionality but rather is a property of geometric cuts themselves. Our counterexample is rather complex, the constructed algorithm trajectory has a fractal nature.