一致凸度量空间中优化的一阶分析：方向次导数与基本下降法
First-Order Analysis of Optimization in Uniformly Convex Metric Spaces: Directional Subderivatives and Basic Descent
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中文总结 AI 辅助
研究一致凸度量空间中函数最小化的一阶方法，通过方向次导数等制定收敛条件，证明存在最速下降方向，为局部光滑非凸优化的一阶显式算法奠定基础。
中文摘要 AI 辅助
我们开发了用于分析和实现一致凸度量空间中最小化函数的显式一阶方法的工具。在包括有界性、测地线光滑性和度量Polyak-Łojasiewicz性质等正则性假设下，根据方向次导数、函数值和迭代来制定下降序列收敛的充分条件。我们证明存在满足收敛假设的最速下降方向。这项工作为局部光滑、非凸优化的一阶显式算法提供了基础。
英文摘要
We develop tools for the analysis and implementation of explicit first-order methods for minimizing functions in uniformly convex metric spaces. We formulate sufficient conditions for convergence of descent sequences in terms of directional subderivatives, function values and iterates under regularity assumptions including boundedness, geodesic smoothness and a metric Polyak-Łojasiewicz property. We show that there exists a steepest descent direction in which the assumptions for convergence are satisfied. This work provides a foundation for first-order explicit algorithms for locally smooth, nonconvex optimization.
