Accelerated Distributed Optimization Algorithm With Malicious Nodes

This paper addresses the challenges of slow convergence and vulnerability to attacks in distributed optimization for multi-agent systems. To improve the convergence rate in the presence of malicious nodes, a novel algorithm that combines the Mean-Subsequence-Reduce (MSR) method with gradient tracking is proposed. This algorithm enables the local variables of normal agents to converge to a convex combination of locally optimal solutions. However, using the MSR method in distributed optimization can lead to a time-varying transition matrix and the loss of the double stochasticity of the transition matrix, which presents design and analysis challenges. To mitigate this issue, a decreasing factor is introduced to alleviate the accumulation of errors caused by the single-row stochasticity of the transition matrix. We prove that the proposed algorithm achieves consensus and convergence to the point satisfying the optimality conditions. Numerical examples demonstrate the effectiveness of the proposed algorithm, showing a faster convergence rate than existing methods.