Continuous-time Proportional-Integral Distributed Optimization for Networked Systems

In this paper we explore the relationship between dual decomposition and the consensus-based method for distributed optimization. The relationship is developed by examining the similarities between the two approaches and their relationship to gradient-based constrained optimization. By formulating each algorithm in continuous-time, it is seen that both approaches use a gradient method for optimization with one using a proportional control term and the other using an integral control term to drive the system to the constraint set. Therefore, a significant contribution of this paper is to combine these methods to develop a continuous-time proportional-integral distributed optimization method. Furthermore, we establish convergence using Lyapunov stability techniques and utilizing properties from the network structure of the multi-agent system.Comment: 23 Pages, submission to Journal of Control and Decision, under review. Takes comments from previous review process into account. Reasons for a continuous approach are given and minor technical details are remedied. Largest revision is reformatting for the Journal of Control and Decisio

A Distributed Asynchronous Method of Multipliers for Constrained Nonconvex Optimization

This paper presents a fully asynchronous and distributed approach for tackling optimization problems in which both the objective function and the constraints may be nonconvex. In the considered network setting each node is active upon triggering of a local timer and has access only to a portion of the objective function and to a subset of the constraints. In the proposed technique, based on the method of multipliers, each node performs, when it wakes up, either a descent step on a local augmented Lagrangian or an ascent step on the local multiplier vector. Nodes realize when to switch from the descent step to the ascent one through an asynchronous distributed logic-AND, which detects when all the nodes have reached a predefined tolerance in the minimization of the augmented Lagrangian. It is shown that the resulting distributed algorithm is equivalent to a block coordinate descent for the minimization of the global augmented Lagrangian. This allows one to extend the properties of the centralized method of multipliers to the considered distributed framework. Two application examples are presented to validate the proposed approach: a distributed source localization problem and the parameter estimation of a neural network.Comment: arXiv admin note: substantial text overlap with arXiv:1803.0648

Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence

In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean to a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.Comment: To appear in IEEE Transactions on Automatic Contro