  We establish the optimal nonergodic sublinear convergence rate of the
proximal point algorithm for maximal monotone inclusion problems. First, the
optimal bound is formulated by the performance estimation framework, resulting
in an infinite dimensional nonconvex optimization problem, which is then
equivalently reformulated as a finite dimensional semidefinite programming
(SDP) problem. By constructing a feasible solution to the dual SDP, we obtain
an upper bound on the optimal nonergodic sublinear rate. Finally, an example in
two dimensional space is constructed to provide a lower bound on the optimal
nonergodic sublinear rate. Since the lower bound provided by the example
matches exactly the upper bound obtained by the dual SDP, we have thus
established the worst case nonergodic sublinear convergence rate which is
optimal in terms of both the order as well as the constants involved. Our
result sharpens the understanding of the fundamental proximal point algorithm.
