提出一种自适应非单调次梯度法，可高效求解非凸非光滑函数优化问题。
Nonmonotone subgradient methods based on a local descent lemma
- 基于局部下降引理设计非单调线搜索策略，适用于上C²类函数。
- 算法收敛到驻点，且在聚类问题中比传统方法更快更稳定。
- 自适应参数更新机制，适合对精度要求高的优化任务。
本文提出一种专为上-$\mathcal{C}^2$ 函数设计的非单调线搜索次梯度算法。这类函数具有非光滑、非凸特性，并满足一种局部化的下降引理，适合进行线搜索。我们证明了所提算法的子序列收敛性，能收敛至优化问题的驻点。该方法涵盖多种次梯度算法，包括牛顿与拟牛顿方法。此外，我们提出了通用框架的一个具体实现——自适应非单调次梯度法（SNSM），可自动更新线搜索参数。特别针对最小平方和聚类问题，给出了SNSM的具体实现。通过数值实验，验证了SNSM相比已有算法在收敛速度和稳定性上的优势。
原文摘要 · Abstract (English)
In this paper we present a nonmonotone line search subgradient algorithm tailored to upper-$\mathcal{C}^2$ functions. This is a family of nonsmooth and nonconvex functions that satisfies a nonsmooth and local version of the descent lemma, making them suitable for line searches. We prove subsequential convergence of the proposed algorithm to a stationary point of the optimization problem. Our approach allows us to cover the setting of various subgradient algorithms, including Newton and quasi-Newton methods. In addition, we propose a specification of the general scheme, named Self-adaptive Nonmonotone Subgradient Method (SNSM), which automatically updates the parameters of the line search. Particular attention is paid to the minimum sum-of-squares clustering problem, for which we provide a concrete implementation of SNSM. We conclude with some numerical experiments where we exhibit the advantages of SNSM in comparison with some known algorithms.
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