  In this paper we consider a distributed optimization scenario in which a set
of processors aims at minimizing the maximum of a collection of "separable
convex functions" subject to local constraints. This set-up is motivated by
peak-demand minimization problems in smart grids. Here, the goal is to minimize
the peak value over a finite horizon with: (i) the demand at each time instant
being the sum of contributions from different devices, and (ii) the local
states at different time instants being coupled through local dynamics. The
min-max structure and the double coupling (through the devices and over the
time horizon) makes this problem challenging in a distributed set-up (e.g.,
well-known distributed dual decomposition approaches cannot be applied). We
propose a distributed algorithm based on the combination of duality methods and
properties from min-max optimization. Specifically, we derive a series of
equivalent problems by introducing ad-hoc slack variables and by going back and
forth from primal and dual formulations. On the resulting problem we apply a
dual subgradient method, which turns out to be a distributed algorithm. We
prove the correctness of the proposed algorithm and show its effectiveness via
numerical computations.
