发表机构
Lehigh University（理海大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文拓展了梯度-本征步算法的分析，证明其在特定条件下具备局部线性收敛速率，还可作为子问题求解器应用于渐进采样策略，以优化大样本等式约束优化的样本复杂度。
AI 中文摘要
为求解非凸等式约束优化问题，Goyens等人提出的近期梯度-本征步算法是一种迭代高效的方法，该算法基于最小化Fletcher增广拉格朗日函数，可从任意起始点寻找近似二阶驻点。本文对该算法的分析进行了拓展，提供了两方面贡献：其一，证明若该算法在充分靠近强二阶驻点处初始化，且采用充分小的步长参数、充分大的惩罚参数，则可获得局部线性收敛速率；此时该算法退化为应用于最小化Fletcher增广拉格朗日的梯度下降算法。其二，作为上述结果的一项特别有用的应用，证明当目标函数和约束函数由大样本均值定义时，梯度-本征步算法可作为迭代高效的子问题求解器，用于求解等式约束优化问题的渐进采样策略，最终与直接求解全样本问题的方法相比，该算法具备更优的最坏情况样本复杂度。
英文摘要
For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.
