  This paper focuses on a class of inclusion problems of maximal monotone
operators in a multi-agent network, where each agent is characterized by an
operator that is not available to any other agents, but the agents can
cooperate by exchanging information with their neighbors according to a given
communication topology. All agents aim at finding a common decision vector that
is the solution to the sum of agents' operators. This class of problems is
motivated by distributed convex optimization with coupled constraints. In this
paper, we propose a distributed proximal point method with a cumulative
correction term (named Proximal-Correction Algorithm) for this class of
inclusion problems of operators. It's proved that the Proximal-Correction
Algorithm converges for any value of a constant penalty parameter. In order to
make the Proximal-Correction ALgorithm computationally implementable for a wide
variety of distributed optimization problems, we adopt two inexact criteria for
calculating the proximal steps of the algorithm. Under each of these two
criteria, the convergence of Proximal-Correction Algorithm can be guaranteed,
and the linear convergence rate is established when the stronger one is
satisfied. In numerical simulations, both exact and inexact versions of
Proximal-Correction Algorithm are executed for a distributed convex
optimization problem with coupled constraints. Compared with several
alternative algorithms in the literature, the exact and inexact versions of
Proximal-Correction both exhibit good numerical performance.
