
Journal of Convex Analysis 23 (2016), No. 1, 237261 Copyright Heldermann Verlag 2016 Proximal Point Algorithm, DouglasRachford Algorithm and Alternating Projections: a Case Study Heinz H. Bauschke Dept. of Mathematics, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada heinz.bauschke@ubc.ca Minh N. Dao Dept. of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam and: Dept. of Mathematics, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada minhdn@hnue.edu.vn Dominikus Noll Institut de Mathématiques, Université de Toulouse, 118 route de Narbonne, 31062 Toulouse, France noll@mip.upstlse.fr Hung M. Phan Dept. of Mathematical Sciences, University of Massachusetts Lowell, 265 Riverside Street, Olney Hall 428, Lowell, MA 01854, U.S.A. hung_phan@uml.edu Many iterative methods for solving optimization or feasibility problems have been invented, and often convergence of the iterates to some solution is proven. Under favourable conditions, one might have additional bounds on the distance of the iterate to the solution leading thus to worst case estimates, i.e., how fast the algorithm must converge. Exact convergence estimates are typically hard to come by. In this paper, we consider the complementary problem of finding best case estimates, i.e., how slow the algorithm has to converge, and we also study exact asymptotic rates of convergence. Our investigation focuses on convex feasibility in the Euclidean plane, where one set is the real axis while the other is the epigraph of a convex function. This case study allows us to obtain various convergence rate results. We focus on the popular method of alternating projections and the DouglasRachford algorithm. These methods are connected to the proximal point algorithm which is also discussed. Our findings suggest that the DouglasRachford algorithm outperforms the method of alternating projections in the absence of constraint qualifications. Various examples illustrate the theory. Keywords: Alternating projections, convex feasibility problem, convex set, DouglasRachford algorithm, projection, proximal mapping, proximal point algorithm, proximity operator. MSC: 65K05; 65K10, 90C25 [ Fulltextpdf (312 KB)] for subscribers only. 