In this work, we introduce the Resilient Projected Push-Pull (RP3) algorithm designed for distributed optimization in multi-agent cyber-physical systems with directed communication graphs and the presence of malicious agents. Our algorithm leverages stochastic inter-agent trust values and gradient tracking to achieve geometric convergence rates in expectation even in adversarial environments. We introduce growing constraint sets to limit the impact of the malicious agents without compromising the geometric convergence rate of the algorithm. We prove that RP3 converges to the nominal optimal solution almost surely and in the $r$-th mean for any $r\geq 1$, provided the step sizes are sufficiently small and the constraint sets are appropriately chosen. We validate our approach with numerical studies on average consensus and multi-robot target tracking problems, demonstrating that RP3 effectively mitigates the impact of malicious agents and achieves the desired geometric convergence.

翻译：本文提出了一种针对具有有向通信图且存在恶意代理的多智能体信息物理系统的分布式优化算法——弹性投影推挽算法。该算法利用智能体间随机信任值与梯度跟踪技术，即使在对抗性环境中也能实现期望意义上的几何收敛速率。我们引入了增长约束集以限制恶意代理的影响，同时不损害算法的几何收敛速率。我们证明，只要步长足够小且约束集选择恰当，RP3算法几乎必然收敛于标称最优解，且对任意$r\geq 1$均具有$r$阶矩收敛性。通过平均共识与多机器人目标跟踪问题的数值研究，我们验证了RP3算法能有效抑制恶意代理的影响并实现预期的几何收敛。