| Summary: | This work investigates a problem of simultaneous global cost minimization and Nash equilibrium seeking, which commonly exists in N-cluster non-cooperative games. Specifically, the players 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 explicit analytical expressions of the players' local cost functions 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 players across different clusters to choose different constant step-sizes. When the largest step-size is sufficiently small, we prove a linear convergence of the players' 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 smoothing parameter. The performance of the proposed algorithm is validated by numerical simulations.
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