In this paper we consider a distributed optimization scenario in which a setof processors aims at minimizing the maximum of a collection of "separableconvex functions" subject to local constraints. This set-up is motivated bypeak-demand minimization problems in smart grids. Here, the goal is to minimizethe peak value over a finite horizon with: (i) the demand at each time instantbeing the sum of contributions from different devices, and (ii) the localstates at different time instants being coupled through local dynamics. Themin-max structure and the double coupling (through the devices and over thetime horizon) makes this problem challenging in a distributed set-up (e.g.,well-known distributed dual decomposition approaches cannot be applied). Wepropose a distributed algorithm based on the combination of duality methods andproperties from min-max optimization. Specifically, we derive a series ofequivalent problems by introducing ad-hoc slack variables and by going back andforth from primal and dual formulations. On the resulting problem we apply adual subgradient method, which turns out to be a distributed algorithm. Weprove the correctness of the proposed algorithm and show its effectiveness vianumerical computations.
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