发表机构
Wayne State University（韦恩州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对非光滑优化中二阶信息利用困难的问题，提出基于余导数的牛顿框架，结合自适应正则化与非单调线搜索，实现全局及局部超线性收敛，并在Lasso、逻辑Lasso和SVM上验证了高效性与鲁棒性。
AI 中文摘要
牛顿型算法因其快速局部收敛性而成为求解优化问题最有效的方法之一。然而，将这些方法扩展到非光滑优化颇具挑战性，原因在于难以纳入二阶信息。为解决这一问题，我们利用变分分析和广义微分工具，提出了一种基于余导数的牛顿框架，用于求解无约束和有约束的非光滑优化问题。所提方法采用广义海森矩阵（定义为次梯度映射的余导数），适用于 $C^{1,1}$ 函数以及具有扩展实值分量的凸复合优化问题。为增强鲁棒性，该算法引入自适应正则化策略，使其能够处理广义海森矩阵为半正定的问题。此外，开发了一种混合自适应非单调（HAN）线搜索方案作为全局化技术，以改善全局收敛性和实际性能。在标准假设下，算法的精确版本和非精确版本均全局收敛，并且当相关次梯度映射满足 semismooth* 性质时，可实现局部超线性收敛。所提框架通过前向后向包络进一步扩展到凸复合优化，使其能够处理目标函数光滑分量具有或不具有强凸性的问题。在Lasso、逻辑Lasso和支持向量机（SVM）问题上的数值实验证明了所提算法的效率和鲁棒性。与几种成熟的一阶和二阶非光滑优化方法的比较表明，所提方法在计算上具有竞争力，同时保持强收敛性质。
英文摘要
Newton-type algorithms are among the most effective methods for solving optimization problems because of their rapid local convergence. However, extending these methods to nonsmooth optimization is challenging due to the difficulty of incorporating second-order information. To address this issue, we propose a coderivative-based Newton framework for solving both unconstrained and constrained nonsmooth optimization problems using tools from variational analysis and generalized differentiation. The proposed method employs generalized Hessians, defined as coderivatives of the subgradient mapping, and is applicable to both $C ^{1,1}$ functions and convex composite optimization problems with extended-real-valued components. To enhance robustness, the algorithm incorporates an adaptive regularization strategy that enables it to handle problems whose generalized Hessians are positive semidefinite. In addition, a Hybrid Adaptive Nonmonotone (HAN) line search scheme is developed as a globalization technique to improve global convergence and practical performance. Under standard assumptions, both the exact and inexact versions of the algorithm are globally convergent and achieve local superlinear convergence when the associated subgradient mapping satisfies the semismooth* property. The proposed framework is further extended to convex composite optimization through the forward-backward envelope, allowing it to handle problems with or without strong convexity in the smooth component of the objective function. Numerical experiments on Lasso, logistic Lasso, and support vector machine (SVM) problems demonstrate the efficiency and robustness of the proposed algorithm. Comparisons with several well-established first-order and second-order methods for nonsmooth optimization show that the proposed approach is computationally competitive while maintaining strong convergence properties.
