Proximal Point Algorithm, Douglas-Rachford Algorithm and Alternating Projections: a Case Study

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 Douglas-Rachford algorithm. These methods are connected to the proximal point algorithm which is also discussed. Our findings suggest that the Douglas-Rachford algorithm outperforms the method of alternating projections in the absence of constraint qualifications. Various examples illustrate the theory.