Extrapolation algorithm for affine-convex feasibility problems Heinz H. Bauschke ∗ , Patrick L. Combettes † , and Serge G. Kruk ‡ October 5, 2005 Abstract The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations. 1 Introduction Let (S i ) i∈I be a countable family of intersecting closed convex sets in a real Hilbert space H. The associated convex feasibility problem is to find x ∈  i∈I S i . (1.1) This problem has a long and rich history in applied mathematics, going back at least to the nineteenth century [13]. We refer the reader to [3, 14, 16, 18, 20, 29] for surveys and background, and to [12] for recent developments. The early methods by Cimmino [17] and by Kaczmarz [41] on systems of linear equations relied on projections. For every i ∈ I , let P i denote the projection operator onto S i . A sequential projection method generates a sequence (x n ) n∈N in H according to the recursion x n+1 = x n + λ n (P i(n) x n − x n ), where i(n) ∈ I and 0 <λ n < 2. (1.2) ∗ Mathematics, Irving K. Barber School, UBC Okanagan, Kelowna, B.C. V1V 1V7, Canada. E-mail: heinz.bauschke@ubc.ca. † Laboratoire Jacques-Louis Lions, Universit´ e Pierre et Marie Curie – Paris 6, 75005 Paris, France. E-mail: plc@math.jussieu.fr. ‡ Department of Mathematics and Statistics, Oakland University, Rochester, Michigan 48309-4485, USA. E-mail: sgkruk@acm.org. 1