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Many iterative methods for solving optimization or feasibility problems havebeen invented, and often convergence of the iterates to some solution isproven. Under favourable conditions, one might have additional bounds on thedistance 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, weconsider the complementary problem of finding best case estimates, i.e., howslow the algorithm has to converge, and we also study exact asymptotic rates ofconvergence. Our investigation focuses on convex feasibility in the Euclideanplane, where one set is the real axis while the other is the epigraph of aconvex function. This case study allows us to obtain various convergence rateresults. We focus on the popular method of alternating projections and theDouglas-Rachford algorithm. These methods are connected to the proximal pointalgorithm which is also discussed. Our findings suggest that theDouglas-Rachford algorithm outperforms the method of alternating projections inthe absence of constraint qualifications. Various examples illustrate thetheory.

Proximal point algorithm, Douglas-Rachford algorithm and alternating projections: a case study