Distributed Unconstrained Optimization with Time-varying Cost Functions

TL;DR

This paper introduces a novel distributed method for tracking the optimal trajectory in time-varying unconstrained optimization problems across networked agents, combining derivative estimation and consensus dynamics.

Contribution

It presents a two-stage approach integrating derivative sampling and consensus to effectively track the optimal solution in dynamic, distributed settings.

Findings

The method achieves asymptotic convergence to the optimal trajectory.

Lyapunov analysis provides an upper bound on the gradient of the total cost.

Numerical simulations demonstrate the impact of parameter tuning on convergence.

Abstract

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.