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We establish the optimal nonergodic sublinear convergence rate of theproximal point algorithm for maximal monotone inclusion problems. First, theoptimal bound is formulated by the performance estimation framework, resultingin an infinite dimensional nonconvex optimization problem, which is thenequivalently reformulated as a finite dimensional semidefinite programming(SDP) problem. By constructing a feasible solution to the dual SDP, we obtainan upper bound on the optimal nonergodic sublinear rate. Finally, an example intwo dimensional space is constructed to provide a lower bound on the optimalnonergodic sublinear rate. Since the lower bound provided by the examplematches exactly the upper bound obtained by the dual SDP, we have thusestablished the worst case nonergodic sublinear convergence rate which isoptimal in terms of both the order as well as the constants involved. Ourresult sharpens the understanding of the fundamental proximal point algorithm.

Optimal Nonergodic Sublinear Convergence Rate of Proximal Point Algorithm for Maximal Monotone Inclusion Problems