A CONJECTURE BY DE PIERRO IS TRUE FOR TRANSLATES OF REGULAR SUBSPACES
Heinz H. Bauschke, Mclean R. Edwards · 2005
Abstract. Suppose we are given nitely many nonempty closed convex sets in a real Hilbert space and their associated projections. For suitable arrangements of the sets, it is known that the sequence obtained by iterating the composi-tion of the underrelaxed projections is weakly convergent. The question arises how these weak limits vary as the underrelaxation parameter tends to zero. In 2001, De Pierro conjectured that the weak limits approach the least squares so-lution nearest to the starting point of the sequence. In fact, a result by Censor, Eggermont, and Gordon implies De Pierro's conjecture for ane subspaces in Euclidean space. This paper extends the result by Censor et al. from Euclidean to Hilbert space. We show that De Pierro's conjecture is true for translates of regular subspaces and the limits all exist with respect to the norm topology. Regularity always holds in Euclidean space. However, this condition is not automatic in innite-dimensional Hilbert space. Two subspaces are constructed to illustrate the possible divergence of the iterates of the composition of the underrelaxed projections. Somewhat surprisingly, examples in the Euclidean plane demonstrate that the approach to the least squares solution can be nonlinear. 1.