Distributed Proximal-Correction Algorithm for the Sum of Maximal Monotone Operators

This article focuses on a class of inclusion problems of maximal monotone operators over directed multiagent networks, where each agent is characterized by a splittable local operator. By integrating the ideas from correction, inexact proximal methods, and push-sum protocol, we proposed a distributed proximal-correction algorithm (referred to as DPCA-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon _{k}^{i}$</tex-math></inline-formula>) to solve this inclusion problem of operators, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon _{k}^{i}$</tex-math></inline-formula> is an agent predetermined accuracy for approximately computing proximal mappings. Under mild assumptions on problem setup and agents specified precision <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon _{k}^{i}$</tex-math></inline-formula>, we proved the convergence and sublinear rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$o(1/k)$</tex-math></inline-formula> of DPCA-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon _{k}^{i}$</tex-math></inline-formula> in terms of the first-order optimality residual. Furthermore, under the metric regularity of local operators of agents, we established its linear convergence rate. Numerical simulations and comparisons with other distributed algorithms illustrate the effectiveness of the proposed algorithm.