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

Abstract

In this paper, we introduce a CBO method that incorporates jumps driven by an $\alpha$-stable stochastic process within a kinetic framework. This approach naturally induces nonlocal stochastic effects, enhancing the exploration capabilities of the method. We formulate the dynamics at the particle level, providing a detailed description of the associated stochastic system and its asymptotic behavior. In particular, by means of a Fourier-based representation, we derive the corresponding fractional Fokker-Planck equation, which captures the nonlocal diffusion mechanisms generated by $\alpha$-stable processes. As a central result, we establish a rigorous convergence analysis for the proposed method. Finally, we assess its performance through a series of numerical experiments, highlighting the effectiveness of $\alpha$-stable jump dynamics and their advantages over standard diffusion based strategies, especially in complex optimization landscapes.