A distributed optimization algorithm over Markov switching topology under adversarial attack

This paper investigates the distributed optimization problem with adversarial agents over Markov switching topology. A Push-DIGing type algorithm is applied to find optimizer of the distributed optimization problem. Different from most existing results on distributed optimization problems with adversary agents, we mainly focus on seeking a fixed step size algorithm under Markov switching communication graph. We first divide the agents into three sets, namely the trust agent set, the normal agent set, and the adversarial agent set. All normal agents only use the state of non-attacked agents for iterations. A practical method is presented to distinguish the adversary agents. It is shown that the states of the trust and normal agents will synchronize to the optimizer with a bounded error if (1) the step size does not exceed an upper bound, (2) the union of the communication graph has a spanning tree, and (3) the differential of a local function is decreasing. Furthermore, we also present the upper and lower bounds on distances between the converged state and the optimizer. An example is presented to verify the effectiveness of our algorithm.