Algorithm Design and Regret Analysis for Aggregative Optimization in Open Multi-Agent Systems

This paper studies the problem of aggregative optimization in open multi-agent systems (OMAS), where agents are allowed to join and leave the system in a free manner during the decision-making process. A multi-aggregator communication mechanism is proposed to facilitate information exchange among agents, in which the aggregators are responsible for collecting information from agents and exchanging them with neighboring aggregators. Based on the multi-aggregator communication mechanism, a novel semi-distributed algorithm is designed. Dynamic regret is taken as a metric to evaluate the performance of the proposed algorithm. It is analytically shown that the dynamic regret can be bounded by the sum of agents' individual regrets, each of which grows sub-linearly with its active period length under locally diminishing step-sizes and a slowly changing environment. Moreover, to quantify the impact of agents' joining and leaving on algorithm performance, the regret is analyzed in the case of agent replacement, wherein some of the agents are replaced by new ones though the number of agents remains constant over time. Additionally, the algorithm is simplified in some special cases, resulting in a tighter regret bound. Finally, two numerical examples on target surrounding problems and price based energy management are given to verify the effectiveness of the proposed methods.