# Dynamic Sharing Through the ADMM

## TL;DR

This paper introduces a dynamic ADMM algorithm for real-time tracking of time-varying optimal solutions in systems with changing costs, demonstrating its convergence and effectiveness through numerical examples.

## Contribution

It proposes a novel dynamic ADMM method for online optimization of time-varying systems, with convergence analysis and practical validation.

## Key findings

- Dynamic ADMM converges linearly to neighborhoods of optimal points.
- The size of neighborhoods depends on the evolution rate of the objective functions.
- Dynamic ADMM performs well in tracking solutions and offers computational advantages.

## Abstract

In this paper, we study a dynamic version of the sharing problem, in which a dynamic system cost function composed of time-variant local costs of subsystems and a shared time-variant cost of the whole system is minimized. A dynamic alternating direction method of multipliers (ADMM) is proposed to track the varying optimal points of the dynamic optimization problem in an online manner. We analyze the convergence properties of the dynamic ADMM and show that, under several standard technical assumptions, the iterations of the dynamic ADMM converge linearly to some neighborhoods of the time-varying optimal points. The sizes of these neighborhoods depend on the drifts of the dynamic objective functions: the more drastically the dynamic objective function evolves across time, the larger the sizes of these neighborhoods. We also investigate the impact of the drifts on the steady state convergence behaviors of the dynamic ADMM. Finally, two numerical examples, namely a dynamic sharing problem and the dynamic least absolute shrinkage and selection operator (LASSO), are presented to corroborate the effectiveness of the proposed dynamic ADMM. It is observed that the dynamic ADMM can track the time-varying optimal points quickly and accurately. For the dynamic LASSO, the dynamic ADMM has competitive performance compared to the benchmark offline optimizor while the former possesses significant computational advantage over the latter.