A Decentralized Second-Order Method for Dynamic Optimization

TL;DR

This paper introduces a decentralized second-order optimization algorithm that effectively tracks time-varying solutions in networked systems with convergence guarantees.

Contribution

It applies the Exact Second-Order Method (ESOM) to dynamic problems, providing a novel decentralized approach with proven linear convergence under certain conditions.

Findings

The proposed method converges linearly to an error bound.

Numerical results show effective tracking of optimal solutions.

The algorithm operates with primal and dual updates on a quadratic approximation.

Abstract

This paper considers decentralized dynamic optimization problems where nodes of a network try to minimize a sequence of time-varying objective functions in a real-time scheme. At each time slot, nodes have access to different summands of an instantaneous global objective function and they are allowed to exchange information only with their neighbors. This paper develops the application of the Exact Second-Order Method (ESOM) to solve the dynamic optimization problem in a decentralized manner. The proposed dynamic ESOM algorithm operates by primal descending and dual ascending on a quadratic approximation of an augmented Lagrangian of the instantaneous consensus optimization problem. The convergence analysis of dynamic ESOM indicates that a Lyapunov function of the sequence of primal and dual errors converges linearly to an error bound when the local functions are strongly convex and have Lipschitz continuous gradients. Numerical results demonstrate the claim that the sequence of iterates generated by the proposed method is able to track the sequence of optimal arguments.