Zero-Inertia Limit: from Particle Swarm Optimization to Consensus Based Optimization

TL;DR

This paper rigorously derives the consensus based optimization (CBO) algorithm from particle swarm optimization (PSO) by taking the zero-inertia limit of the stochastic differential equations, providing convergence rates and numerical validation.

Contribution

It provides the first rigorous derivation of CBO from PSO via zero-inertia limit, including convergence analysis and numerical illustrations.

Findings

Established a formal link between PSO and CBO through zero-inertia limit.

Derived a quantified convergence rate for the limit process.

Validated theoretical results with numerical examples.

Abstract

Recently a continuous description of the particle swarm optimization (PSO) based on a system of stochastic differential equations was proposed by Grassi and Pareschi in arXiv:2012.05613 where the authors formally showed the link between PSO and the consensus based optimization (CBO) through zero-inertia limit. This paper is devoted to solving this theoretical open problem proposed in arXiv:2012.05613 by providing a rigorous derivation of CBO from PSO through the limit of zero inertia, and a quantified convergence rate is obtained as well. The proofs are based on a probabilistic approach by investigating the weak convergence of the corresponding stochastic differential equations (SDEs) of Mckean type in the continuous path space and the results are illustrated with some numerical examples.