A proximal point algorithm converging strongly for general errors

In this paper a proximal point algorithm (PPA) for maximal monotone operators with appropriate regularization parameters is considered. A strong convergence result for PPA is stated and proved under the general condition that the error sequence tends to zero in norm. Note that Rockafellar (SIAM J Control Optim 14:877–898, 1976) assumed summability for the error sequence to derive weak convergence of PPA in its initial form, and this restrictive condition on errors has been extensively used sofar for different versions of PPA. Thus this Note provides a lutiontoalongstandingopenproblemandinparticularoffersnewpossibilitiestowards
the approximation of the minimum points of convex functionals.