# A duality-based approach for distributed min-max optimization with application to demand side management

## TL;DR

This paper introduces a novel distributed algorithm for min-max optimization problems with applications to demand side management in smart grids, addressing challenges of coupling and structure that hinder traditional methods.

## Contribution

It develops a duality-based distributed approach for complex min-max problems with coupled constraints, not solvable by standard dual decomposition techniques.

## Key findings

- Algorithm is proven correct and convergent.
- Numerical results demonstrate effectiveness in demand management.
- Addresses challenges of double coupling in distributed optimization.

## Abstract

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.