AI 中文总结
本文提出用于非凸优化的自适应Barzilai-Borwein邻近梯度方法（AdaBBNC），将BB步长策略扩展至复合非凸优化问题，证明其迭代复杂度为O(ε⁻²)，实验显示其在病态问题中表现更优。
AI 中文摘要
Barzilai-Borwein（BB）方法是一种高效的基于梯度的无约束优化方法，可近似Hessian矩阵的谱信息，以较低计算成本捕捉曲率。本文将BB步长策略扩展至由光滑非凸项与正常闭凸项组成的复合非凸优化问题，提出用于非凸优化的自适应Barzilai-Borwein邻近梯度方法（AdaBBNC）。该方法将灵活的基于BB的曲率估计融入邻近梯度框架，以提升非凸场景下的适应性。在温和假设下，证明AdaBBNC在寻找ε-驻点时达到最优迭代复杂度O(ε⁻²)，无需全局Lipschitz常数的先验知识。数值实验验证了该方法的有效性与鲁棒性，与近期无参数、无线搜索的自适应邻近梯度方法相比，AdaBBNC在病态优化问题中表现出更具进取性且稳定的行为。
英文摘要
The Barzilai-Borwein (BB) method is an efficient gradient-based approach for unconstrained optimization that approximates spectral information of the Hessian matrix to capture curvature at low computational cost. In this paper, we extend the BB stepsize strategy to composite nonconvex optimization problems consisting of a smooth nonconvex term and a proper closed convex term, and propose an adaptive Barzilai-Borwein proximal gradient method for nonconvex optimization (AdaBBNC). The proposed method incorporates a flexible BB-based curvature estimate into the proximal gradient framework to enhance adaptability in nonconvex settings. Under mild assumptions, we establish that AdaBBNC achieves the optimal iteration complexity of $\mathcal{O}(ε^{-2})$ for finding an $ε$-stationary point, without requiring any prior knowledge of the global Lipschitz constant. Numerical experiments demonstrate the effectiveness and robustness of the proposed method. Compared with recent parameter-free and line-search-free adaptive proximal gradient methods, AdaBBNC exhibits more aggressive yet stable behavior in ill-conditioned optimization problems.
