Distributed Multi-Agent Optimization Based on a Constrained Subgradient Method

Abstract

This paper proposes a protocol for a distributed optimization problem to minimize the average of objective functions of the agents in the network with satisfying constraints of each agent. The protocol can handle uncommon constraints of the agents. Instead of invoking dual functions, only 1-bit information on fulfillment of the constraint of each agent is transmitted between agents as well as the decision variable. The proof of consensus and convergence is provided based on the constrained subgradient method. A numerical example illustrates how the proposed protocol works.

Simple Synchronous and Asynchronous Algorithms for Distributed Minimax Optimization

“…12, 14, 16, and 18). The convergence speed of the synchronous algorithm with the scalable step size (13) are better than the asynchronous one of the simple diminishing step size (12), but worse than the one with the scalable step size (13). Thus, we see that the scalable diminishing step size (7) and (13) are effective to enhance the convergence speed of the algorithms.…”

“…5 shows that the result of p = 1 by the asynchronous algorithm with the simple diminishing step size (12), and Fig. 6 shows that the result of p = 1 with the scalable step size (13). Note that the optimal solution for this instance in (3) (p = 1) is 1.00.…”

“…Notice here that, in order to get the optimal solution of the original minimax optimization, since all of the nodes have to evaluate the maximum value over the functions, the nodes must exchange their values of the functions as well as their decision variables in principle. In the existing algorithms to solve the original minimax optimization [12], [13], each node has to exchange further information in addition to the decision variable. Otherwise, we have to employ a non-standard distributed algorithm [14].…”

Distributed Multi-Agent Optimization Based on an Exact Penalty Method with Equality and Inequality Constraints

“…We have shown (9) by Lemma 2. To prove (10) and (12), we assume the following, which would lead to a contradiction: there exist ε > 0 and integer t 1 such that for all t ≥ t 1 it holds that…”

“…Protocols shown there involve optimization along with updates of dual variables. On the other hand, the authors have shown approaches via a protocol with additional information on fulfillment of constraints [10], [11], where the protocol works with decision variables and small portion of data that indicates fulfillment of the local constraints of agents.…”