The Developments of Proximal Point Algorithms

The problem of finding a zero point of a maximal monotone operator plays a central role in modeling many application problems arising from various fields, and the proximal point algorithm (PPA) is among the fundamental algorithms for solving the zero-finding problem. PPA not only provides a very general framework of analyzing convergence and rate of convergence of many algorithms, but also can be very efficient in solving some structured problems. In this paper, we give a survey on the developments of PPA and its variants, including the recent results with linear proximal term, with the nonlinear proximal term, as well as the inexact forms with various approximate criteria.