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This paper presents a randomized gradient-free distributed online optimization algorithm, with a group of agents whose local objective functions are time-varying. It is worth noting that the value of the local objective function is only revealed to the corresponding agent after the decision is made at each time-step. Thus, each agent updates the decision variable using the local objective function value of its last decision and the information collected from its immediate in-neighbors. A randomized gradient-free oracle is built locally in replacement of the true gradient information in guiding the updates of the decision variable. The notion of dynamic regret is brought forward to measure the difference between the total cost incurred by the agent's state estimation and the offline centralized optimal solution where the objective functions are available a priori. Under the assumptions of strongly connected communication graph and bounded subgradients of the local objective functions, we characterize the dynamic regret associated with each agent as a function of the time duration T and the deviation of the minimizer sequence. Averaging the dynamic regret over the time duration, we establish the asymptotic convergence to a small neighborhood of zero with a rate of O(ln T / √ T ). The effectiveness of this algorithm is illustrated through numerical simulations.

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In this paper, we consider a problem of simultaneous global cost minimization and Nash equilibrium seeking, which commonly exists in N -cluster non-cooperative games. Specifically, the agents in the same cluster collaborate to minimize a global cost function, being a summation of their individual cost functions, and jointly play a non-cooperative game with other clusters as players. For the problem settings, we suppose that the agents' local cost functions can be non-smooth, the explicit analytical expressions are unknown, but the function values can be measured. We propose a gradient-free Nash equilibrium seeking algorithm by a synthesis of Gaussian smoothing techniques and gradient tracking. Furthermore, instead of using the uniform coordinated step-size, we allow the agents to choose their own specific constant step-sizes. When the largest step-size is sufficiently small, we prove a linear convergence of the agents' actions to a neighborhood of the unique Nash equilibrium under a strongly monotone game mapping condition, with the error gap being propotional to the largest step-size and the heterogeneity of the step-size. Moreover, we provide both upper and lower bounds for the rate of convergence of the proposed algorithm. The performance of the proposed algorithm is validated by numerical simulations.

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