Consensus-based optimization with $\alpha$-stable jump processes

Abstract

In this paper, we introduce a novel variant of the CBO method that incorporates jumps according to an $\alpha$-stable stochastic process in a kinetic framework. This extension gives rise to nonlocal stochastic effects, which improve the exploration capabilities of the method. We formulate the method at the particle level, detailing the corresponding stochastic dynamics and its asymptotic behavior. In particular, through a Fourier-based representation, we derive the associated fractional Fokker-Planck equation, which naturally accounts for the nonlocal diffusion behaviors induced by $\alpha$-stable processes. As a central result, we establish a rigorous convergence result for the proposed approach. Finally, we evaluate the performance of the method through a set of numerical experiments. The results demonstrate the effectiveness of the $\alpha$-stable jump process and emphasize its potential advantages over standard diffusion-based methods, particularly in complex optimization settings.