Robust decentralized dynamic optimization at presence of malfunctioning agents

This paper considers the problem of tracking a network-wide solution that dynamically minimizes the summation of time-varying local cost functions of network agents, when some of the agents are malfunctioning. The malfunctioning agents broadcast faulty values to their neighbors, and lead the optimization process to a wrong direction. To mitigate the influence of the malfunctioning agents, we propose a total variation (TV) norm regularized formulation that drives the local variables of the regular agents to be close, while allows them to be different with the faulty values broadcast by the malfunctioning agents. We give a sufficient condition under which consensus of the regular agents is guaranteed, and bound the gap between the consensual solution and the optimal solution we pursue as if the malfunctioning agents do not exist. A fully decentralized subgradient algorithm is proposed to solve the TV norm regularized problem in a dynamic manner. At every time, every regular agent only needs one subgradient evaluation of its current local cost function, in addition to combining messages received from neighboring regular and malfunctioning agents. The tracking error is proved to be bounded, given that variation of the optimal solution is bounded. Numerical experiments demonstrate the robust tracking performance of the proposed algorithm at presence of the malfunctioning agents.