Abstract
In this paper, we study a dynamic version of the sharing problem, in which a dynamic system cost function composed of time-variant local costs of subsystems and a shared time-variant cost of the whole system is minimized. A dynamic alternating direction method of multipliers (ADMM) is proposed to track the varying optimal points of the dynamic optimization problem in an online manner. We analyze the convergence properties of the dynamic ADMM and show that, under several standard technical assumptions, the iterations of the dynamic ADMM converge linearly to some neighborhoods of the time-varying optimal points. The sizes of these neighborhoods depend on the drifts of the dynamic objective functions: the more drastically the dynamic objective function evolves across time, the larger the sizes of these neighborhoods. We also upper bound the limiting optimality gaps of the dynamic ADMM explicitly, and analyze its regret and constraint violation. Finally, two numerical examples are presented to corroborate the effectiveness of the proposed dynamic ADMM.
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