发表机构
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对多变量非光滑目标的极值寻优问题，通过修改扰动-解调架构，利用一般平均理论推导了其全局稳定性条件，并通过数值示例验证了方法的有效性。
AI 中文摘要
经典极值寻优（ES）通常被解释为梯度下降的近似，但该解释在连续时间多变量设置下的非光滑目标中不够清晰。我们对经典多变量扰动-解调架构提出了最小修改：采用有理独立的扰动频率及匹配的解调信号。对于任意局部利普希茨静态目标，克罗内克-外尔定理表明，在每个固定扰动振幅下，长时间平均动力学恰好是核平滑目标的负梯度。由于有理独立频率使扰动和解调信号非周期，我们采用一般平均理论而非周期平均理论。若平滑目标的梯度流是全局一致渐近稳定的，则ES动力学实际为全局一致渐近稳定。我们还推导了关联扰动占据密度、解调信号和平滑核的一般匹配条件，得到一系列替代设计。数值示例包括可解释为非线性规划惩罚函数的非光滑目标函数，以及Rastrigin函数，其平滑操作消除了所有不必要的局部极小值。
英文摘要
Classical extremum seeking (ES) is commonly interpreted as approximating gradient descent, but this interpretation is less clear for nonsmooth objectives in the continuous-time multivariable setting. We propose a minimal modification of the classical multivariable perturbation--demodulation architecture: rationally independent perturbation frequencies and matched demodulation signals. For any locally Lipschitz static objective, the Kronecker--Weyl theorem shows that, at every fixed perturbation amplitude, the long-time averaged dynamics are exactly the negative gradient of a kernel-smoothed objective. Because rationally independent frequencies render the perturbation and demodulation signals nonperiodic, we employ general averaging theory rather than periodic averaging theory. If the gradient flow of the smoothed objective is globally uniformly asymptotically stable, then the ES dynamics are practically globally uniformly asymptotically stable. We also derive a general matching condition relating the perturbation occupation density, demodulation signal, and smoothing kernel, yielding a family of alternative designs. Numerical examples include a nonsmooth objective function, which may be interpreted as the penalty function of a nonlinear program, and the Rastrigin function, for which smoothing eliminates all undesired local minima.
