  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.
