非凸非光滑函数和的梯度映射型平稳性测度的渐近分析及在邻近梯度型算法中的应用
Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms
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中文总结 AI 辅助
本文针对非凸非光滑函数和提出梯度映射型平稳性测度并做渐近分析，将其应用于邻近梯度算法，解决了一个开放问题并提出带非单调线搜索的邻近变量平滑算法。
中文摘要 AI 辅助
针对两个可能非光滑的非凸函数之和，我们提出了一种梯度映射型平稳性测度。在合适的正则性假设下，我们证明了Fréchet平稳性或邻近平稳性可通过该测度沿某收敛序列的渐近消失来刻画。我们还为该测度的一种基于平滑的变体建立了类似的渐近结果，这使得我们能将所提框架与为非光滑优化开发的平滑技术相结合。作为该平稳性测度分析的一项应用，我们对[Olikier-Waldspurger, SIAM J. Optim., 2025]提出的一个开放问题给出了肯定回答，该问题是：在代价函数的一个分量满足局部Lipschitz光滑性的条件下，邻近梯度方法生成的序列的每个聚点是否为邻近平稳点。作为第二项应用，我们提出了一种带非单调线搜索的邻近变量平滑算法，用于在一个分量正则性较低且另一个分量满足 prox-正则性的条件下最小化两个非光滑非凸函数之和。对于所提算法，我们证明了，使得平稳性测度消失的子序列的每个聚点都是Fréchet平稳点。
英文摘要
We propose a gradient mapping-type stationarity measure for the sum of two possibly nonsmooth nonconvex functions. Under suitable regularity assumptions, we show that Fréchet or proximal stationarity can be characterized through asymptotic vanishing of the proposed measure along some convergent sequence. We also establish analogous asymptotic results for a smoothing-based variant of the measure, which enables us to combine the proposed framework with smoothing techniques developed for nonsmooth optimization. As an application of our analysis of the stationarity measure, we provide an affirmative answer to an open question raised by [Olikier-Waldspurger, SIAM J. Optim., 2025] on whether every cluster point of a sequence generated by a proximal gradient method is a proximal stationary point under local Lipschitz smoothness of one component of the cost function. As a second application, we propose a proximal variable smoothing algorithm with a nonmonotone linesearch for minimizing the sum of two nonsmooth nonconvex functions under lower regularity of one component and prox-regularity of the other. For the proposed algorithm, we show that every cluster point of a subsequence such that the stationarity measure vanishes is a Fréchet stationary point.
发表机构
- Institute of Science Tokyo（东京科学大学）
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