Proximal point algorithm, Douglas-Rachford algorithm and alternating projections: a case study (Q2814277)

The subject of the article is located in a scientifically active area called iterative methods. It usually means a deep analysis related with convergence of the iterates, i.e., how fast the algorithm must converge.NEWLINENEWLINEThe authors investigate three algorithms, the proximal point algorithm (PPA), the method of alternating projections and the Dougles-Rachford algorithm. They consider the complementary problem of finding best case estimates, i.e., how slow the algorithm has to converge, instead of a worst case analysis which is related with additional bounds on the distance of the iterate to the solution. They also study exact asymptotic rates of convergence and obtain various convergence rate results by focusing on convex feasibility in the Euclidean plane. They show that the Dougles-Rachford algorithm outperforms the method of alternating projections in the absence of constraint qualifications.NEWLINENEWLINEThis article is well written, structured and explained, it contains five sections: Section 1 as the introduction, Section 2 on Auxiliary results, Section 3 on Proximal point algorithm, Section 4 on Method of alternating projections, and Section 5 on Douglas-Rachford algorithm.NEWLINENEWLINEIn fact, future scientific work in higher-dimensional space and other classes of convext sets would be interesting.