A Duality-Based Approach for Distributed Min–Max Optimization

In this paper, we consider a distributed optimization scenario, motivated by peak-demand minimization, in which a set of processors aims at cooperatively solving a class of min-max optimization problems. The min-max structure and a twofold coupling make the problem challenging in a distributed setup. We propose a distributed algorithm based on the derivation of a series of dual problems and the application of properties from min-max optimization. The resulting distributed algorithm, despite its complex derivation, has a simple structure consisting of a primal optimization and a suitable dual update. We prove the convergence of the proposed algorithm in objective value, and, moreover, that every limit point of the primal sequence is an optimal (thus feasible) solution. This primal recovery property is of key importance in applications since it allows each agent to compute its portion of the global optimal strategy without resorting to any recovery mechanism. Finally, we provide numerical computations for peak-demand optimization in a network of thermostatically controlled loads and show that our algorithm outperforms a plain distributed subgradient performed on the dual.