In this paper, we consider a continuous description based on stochastic differential equations of the popular particle swarm optimization (PSO) process for solving global optimization problems and derive in the large particle limit the corresponding mean-field approximation based on Vlasov–Fokker–Planck-type equations. The disadvantage of memory effects induced by the need to store the local best position is overcome by the introduction of an additional differential equation describing the evolution of the local best. A regularization process for the global best permits to formally derive the respective mean-field description. Subsequently, in the small inertia limit, we compute the related macroscopic hydrodynamic equations that clarify the link with the recently introduced consensus based optimization (CBO) methods. Several numerical examples illustrate the mean field process, the small inertia limit and the potential of this general class of global optimization methods.

中文翻译：

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从粒子群优化到基于共识的优化：随机建模和平均场极限

在本文中，我们考虑基于流行粒子群优化 (PSO) 过程的随机微分方程的连续描述，用于解决全局优化问题，并在大粒子极限下推导出基于 Vlasov-Fokker-Planck- 的相应平均场近似。类型方程。由需要存储局部最佳位置引起的记忆效应的缺点通过引入描述局部最佳演变的附加微分方程来克服。全局最佳的正则化过程允许正式导出相应的平均场描述。随后，在小惯性极限中，我们计算了相关的宏观流体动力学方程，阐明了与最近引入的基于共识的优化 (CBO) 方法的联系。