Population-based and distributional optimization methods, from evolution strategies and consensus-based optimization to covariance-matrix adaptation and stochastic gradient methods viewed as distributional dynamics, are widely used for nonconvex or black-box problems, yet their convergence analyses remain fragmented across algorithm-specific techniques. We introduce an operator calculus in which a broad class of such methods, after choosing an appropriate state space and, where necessary, augmenting the state by memory or strategy variables, is described as a composition of three elementary operators (mutation, selection, and recombination) acting on probability measures. Under explicit stability and regularity conditions, the composite operator admits a pre-generator whose continuous-time limit is a transport-reaction-jump (TRJ) PDE that preserves the operator splitting. On this foundation we establish a modular Lyapunov principle. If a state-space Lyapunov function both dissipates under the full generator and controls the relevant search-space gauges, then the state-space Lyapunov functional and the induced search errors decay exponentially. The additive generator structure allows dissipation estimates to be assembled operator by operator, providing a toolkit for certifying convergence of composite mean-field algorithms.

翻译：基于群体和分布的优化方法，从进化策略、基于共识的优化到协方差矩阵自适应和作为分布动力学处理的随机梯度方法，广泛应用于非凸或黑箱问题，然而其收敛性分析仍分散于针对特定算法的技术中。本文引入一种算子微积分，通过这种微积分，一大类此类方法（在选取恰当的状态空间并必要时通过记忆或策略变量增强状态后）可描述为作用于概率测度的三个基本算子（变异、选择和重组）的复合。在明确的稳定性与正则性条件下，复合算子允许一个预生成元，其连续时间极限是保留算子分裂的输运-反应-跳跃偏微分方程。在此基础之上，我们建立了一个模块化的李雅普诺夫原理。如果一个状态-空间李雅普诺夫函数在完整生成元下耗散，并控制相关的搜索空间度量，那么状态-空间李雅普诺夫泛函及由此导出的搜索误差以指数速率衰减。可加生成元结构使得耗散估计可以逐个算子进行组装，从而为证明复合平均场算法的收敛性提供了一个工具集。