AI 中文总结
研究粒子群优化算法的长期稳定性与收敛性，通过连接经典稳定性分析与平均场方法，研究无记忆正则化PSO模型，确定其稳定及收敛条件，通过研究线性化动力学、分析非线性平均场系统及建立误差界来实现。
AI 中文摘要
粒子群优化（PSO）是一种全局优化算法，由在搜索空间中演化的相互作用粒子集定义。出于启发式动机，由于其动力学的二阶、随机和高度非线性性质，其理论分析仍然有限。在本文中，我们将停滞假设下的经典PSO稳定性分析与更新的平均场方法联系起来，为时间离散算法提供了新的定量估计。我们特别研究了一个无记忆的正则化PSO模型，通过向原始模型添加噪声底来产生非退化噪声。研究这样一个替代模型使我们能够确定动力学稳定并收敛到全局极小值附近小邻域的定量条件。我们通过首先研究线性化动力学的舒尔稳定性，然后通过拉普拉斯原理分析非线性平均场系统的收敛性质，最后为阶数为\(N^{-1/2}\)的平均场近似建立定量误差界来做到这一点。
英文摘要
Particle Swarm Optimization (PSO) is a global optimization algorithm defined by an interacting set of particles evolving over the search space. Heuristically motivated, its theoretical analysis remains limited due to the second-order, stochastic, and highly nonlinear nature of the dynamics. In this paper, we connect classical PSO stability analysis under the stagnation assumption with more recent mean-field methods, providing new quantitative estimates for the time-discrete algorithm. We study in particular a regularized PSO model without memory, with non-degenerate noise by adding a noise floor to the original model. Studying such a surrogate model allows us to identify quantitative conditions under which the dynamics is stable and converges toward a small neighborhood of a global minimizer. We do so by first studying the Schur stability of the linearized dynamics, then analyzing the convergence properties of a nonlinear mean-field system via a Laplace principle, and finally establishing a quantitative error bound for the mean-field approximation of order $N^{-1/2}$.
