We address the issue of global convergence in stochastic continuous optimization. For that purpose, we formulate the Canonical Evolutionary Strategy (CES) as a controlled mathematical framework to analyze global convergence in evolutionary algorithms via the semiclassical limit of a Schr{ö}dinger-type replicator-mutator equation. We provide a rigorous hierarchy from a discrete individual-based dynamics to a deterministic mean-field limit, demonstrating that global convergence is governed by the principal eigenfunction of the underlying operator. This property, defined as Geometric Selection, naturally prioritizes robust, flat optima over narrow local traps, offering a mathematical justification for the ''survival of the flattest'' phenomenon. Moreover, unlike consensus-driven methods that are prone to premature variance collapse when the global minimizer resides outside the initial support, the replicator-mutator dynamics of CES facilitate intrinsic mass transport. High-dimensional benchmarks (d = 30) confirm this advantage, showing that CES achieves lower residual errors in shifted initialization scenarios where standard consensus-driven and gradient-based methods fail to migrate effectively. By shifting the focus from point-wise consensus to spectral concentration, our framework provides a robust theoretical foundation for global convergence in Evolution Strategies (ES) without the need for additional numerical heuristics.

翻译：我们解决了随机连续优化中的全局收敛问题。为此，我们将规范进化策略（CES）构建为一个受控的数学框架，通过薛定谔型复制子-突变子的半经典极限来分析进化算法的全局收敛性。我们从离散的基于个体的动力学出发，严格推导出确定性平均场极限的层次结构，证明全局收敛由底层算子的主特征函数所支配。这一性质被称为几何选择，自然地优先考虑稳健平坦的最优解而非狭窄的局部陷阱，从而为“最平坦者生存”现象提供了数学依据。此外，与易陷入过早方差崩溃（当全局最小值位于初始支撑集之外时）的共识驱动方法不同，CES的复制子-突变子动力学促进了固有的质量输运。高维基准测试（d=30）证实了这一优势，表明在标准共识驱动和基于梯度的方法无法有效迁移的偏移初始化场景中，CES实现了更低的残差误差。通过将焦点从逐点共识转向谱集中，我们的框架为进化策略（ES）的全局收敛提供了稳健的理论基础，无需额外数值启发式方法。