Decentralized Proximal Method of Multipliers for Convex Optimization with Coupled Constraints

In this paper, a decentralized proximal method of multipliers (DPMM) is proposed to solve constrained convex optimization problems over multi-agent networks, where the local objective of each agent is a general closed convex function, and the constraints are coupled equalities and inequalities. This algorithm strategically integrates the dual decomposition method and the proximal point algorithm. One advantage of DPMM is that subproblems can be solved inexactly and in parallel by agents at each iteration, which relaxes the restriction of requiring exact solutions to subproblems in many distributed constrained optimization algorithms. We show that the first-order optimality residual of the proposed algorithm decays to $0$ at a rate of $o(1/k)$ under general convexity. Furthermore, if a structural assumption for the considered optimization problem is satisfied, the sequence generated by DPMM converges linearly to an optimal solution. In numerical simulations, we compare DPMM with several existing algorithms using two examples to demonstrate its effectiveness

Push-Pull Based Distributed Primal-Dual Algorithm for Coupled Constrained Convex Optimization in Multi-Agent Networks

This paper focuses on a distributed coupled constrained convex optimization problem over directed unbalanced and time-varying multi-agent networks, where the global objective function is the sum of all agents' private local objective functions, and decisions of all agents are subject to coupled equality and inequality constraints and a compact convex subset. In the multi-agent networks, each agent exchanges information with other neighboring agents. Finally, all agents reach a consensus on decisions, meanwhile achieving the goal of minimizing the global objective function under the given constraint conditions. For the purpose of protecting the information privacy of each agent, we first establish the saddle point problem of the constrained convex optimization problem considered in this article, then based on the push-pull method, develop a distributed primal-dual algorithm to solve the dual problem. Under Slater's condition, we will show that the sequence of points generated by the proposed algorithm converges to a saddle point of the Lagrange function. Moreover, we analyze the iteration complexity of the algorithm