Superlinear Drift in Consensus-Based Optimization with Condensation Phenomena

TL;DR

This paper introduces a novel consensus-based optimization method inspired by quantum boson models, exhibiting condensation phenomena and superlinear drift, with theoretical analysis and numerical experiments demonstrating its potential advantages.

Contribution

The paper develops a new CBO algorithm with superlinear drift inspired by boson condensation models, including a mean-field formulation and a marginal approach for higher dimensions.

Findings

The new CBO method exhibits condensation-like phenomena and finite-time blow-up.

Numerical experiments show improved performance over classical CBO.

The marginal formulation effectively extends one-dimensional results to multiple dimensions.

Abstract

Consensus-based optimization (CBO) is a class of metaheuristic algorithms designed for global optimization problems. In the many-particle limit, classical CBO dynamics can be rigorously connected to mean-field equations that ensure convergence toward global minimizers under suitable conditions. In this work, we draw inspiration from recent extensions of the Kaniadakis--Quarati model for indistinguishable bosons to develop a novel CBO method governed by a system of SDEs with superlinear drift and nonconstant diffusion. The resulting mean-field formulation in one dimension exhibits condensation-like phenomena, including finite-time blow-up and loss of $L^2$-regularity. To avoid the curse of dimensionality a marginal based formulation which permits to leverage the one-dimensional results to multiple dimensions is proposed. We support our approach with numerical experiments that highlight both its consistency and potential performance improvements compared to classical CBO methods.