# Distributed Unconstrained Optimization with Time-varying Cost Functions

arXiv: https://arxiv.org/abs/2212.09472 · HTML: https://mlsanity.com/paper/2212.09472/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 30 references that are in the map.
- First posted on arXiv 12 Dec 2022.

## Builds on (most cited first, 9 of 9)

- [Distributed optimization over time-varying directed graphs](https://mlsanity.com/paper/1303.2289/) (2013)
- [A Class of Prediction-Correction Methods for Time-Varying Convex Optimization](https://mlsanity.com/paper/1509.05196/) (2015)
- [Distributed Convex Optimization for Continuous-Time Dynamics with Time-Varying Cost Function](https://mlsanity.com/paper/1507.04878/) (2015)
- [Prediction-Correction Interior-Point Method for Time-Varying Convex Optimization](https://mlsanity.com/paper/1608.07544/) (2016)
- [An Accelerated Method For Decentralized Distributed Stochastic Optimization Over Time-Varying Graphs](https://mlsanity.com/paper/2103.15598/) (2021)
- [Distributed Optimization over Time-varying Graphs with Imperfect Sharing of Information](https://mlsanity.com/paper/2106.08469/) (2021)
- [Dual Prediction-Correction Methods for Linearly Constrained Time-Varying Convex Programs](https://mlsanity.com/paper/1709.05850/) (2017)
- [Continuous-time Proportional-Integral Distributed Optimization for Networked Systems](https://mlsanity.com/paper/1309.6613/) (2013)
- [ADMM for Distributed Dynamic Beamforming](https://mlsanity.com/paper/1609.02784/) (2016)

## Abstract (arXiv)

In this paper, we propose a novel solution for the distributed unconstrained optimization problem where the total cost is the summation of time-varying local cost functions of a group networked agents. The objective is to track the optimal trajectory that minimizes the total cost at each time instant. Our approach consists of a two-stage dynamics, where the first one samples the first and second derivatives of the local costs periodically to construct an estimate of the descent direction towards the optimal trajectory, and the second one uses this estimate and a consensus term to drive local states towards the time-varying solution while reaching consensus. The first part is carried out by the implementation of a weighted average consensus algorithm in the discrete-time framework and the second part is performed with a continuous-time dynamics. Using the Lyapunov stability analysis, an upper bound on the gradient of the total cost is obtained which is asymptotically reached. This bound is characterized by the properties of the local costs. To demonstrate the performance of the proposed method, a numerical example is conducted that studies tuning the algorithm's parameters and their effects on the convergence of local states to the optimal trajectory.