AI 中文总结
针对非凸优化问题，提出通用线搜索牛顿框架，避免重复海森正则化，在温和假设下建立全局收敛性，适当条件下证明局部收敛性，应用于强拟凸优化并给出算法及分析，数值实验验证方法有效性。
AI 中文摘要
牛顿法因其快速的局部收敛性，是光滑优化中最有效的二阶算法之一。然而，现有的全局收敛牛顿型方法通常要求目标函数为凸或强凸，而非凸优化方法往往每次迭代都依赖海森正则化。本文提出了一种用于无约束优化的通用线搜索牛顿框架，仅在牛顿方向定义良好且合适时利用它，避免重复海森正则化。该框架包含几种现有混合梯度 - 牛顿方法作为特殊情况，并自然产生一种新的外梯度牛顿方法。我们在温和假设下建立了全局收敛性，包括Polyak - Lojasiewicz - Kurdyka（PLK）条件，允许孤立和非孤立聚点。在适当的正则性假设下进一步证明了局部超线性和二次收敛性。最后，将该框架应用于强拟凸优化，并提供了据我们所知的首个针对此类重要非凸优化问题的牛顿型算法及全面收敛分析。数值实验证明了所提方法的有效性。
英文摘要
Newton's method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient--Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak--Lojasiewicz--Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.
