A duality-based approach for distributed min-max optimization

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.