AI 中文总结
该研究引入戈德斯坦二阶δ-次微分等概念，针对连续可微函数提出基于立方正则化和高斯平滑同伦的零阶算法，用于寻找近似二阶平稳点，并推导了迭代复杂度。
AI 中文摘要
我们引入了一种新的广义海森矩阵，称为戈德斯坦二阶δ-次微分，以及具有局部利普希茨梯度的连续可微函数的(ε₁,ε₂,δ)-二阶平稳点的相关概念。我们提出了一种基于立方正则化和高斯平滑同伦的零阶算法，以找到利普希茨可微函数的此类近似二阶平稳点，并在目标函数的温和强制型假设下推导了迭代复杂度。
英文摘要
We introduce a new generalized Hessian, called the Goldstein second-order $δ$-subdifferential, and an associated notion of $(ε_1,ε_2,δ)$-second-order stationary point for continuously differentiable functions with locally Lipschitz gradients. We propose a zeroth-order algorithm based on cubic regularization and Gaussian smoothing with homotopy to find such approximate second-order stationary points for Lipschitz differentiable functions, and derive the iteration complexity under a mild coercivity-type assumption on the objective function.
