  In this paper we consider a distributed optimization scenario in which a set
of processors aims at cooperatively solving a class of min-max optimization
problems. 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 device states at different time instants being
coupled through local constraints (e.g., the 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., existing 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 repeatedly apply duality theory and
properly introduce ad-hoc slack variables in order to derive a series of
equivalent problems. On the resulting problem we apply a dual subgradient
method, which turns out to be a distributed algorithm consisting of a
minimization on the original primal variables and a suitable dual update. We
prove the convergence of the proposed algorithm in objective value. Moreover,
we show that every limit point of the primal sequence is an optimal (feasible)
solution. Finally, we provide numerical computations for a peak-demand
optimization problem in a network of thermostatically controlled loads.
