发表机构
Federal University of Goias（戈亚斯联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究Hadamard流形上闭测地凸可行集的约束光滑优化问题，分析固定步长与回溯线搜索两种投影梯度方案，证明聚点的一阶平稳性，推导迭代复杂度界，并通过对称正定矩阵流形上的约束Karcher均值问题验证方法性能。
AI 中文摘要
我们研究可行集为闭测地凸集的Hadamard流形上的约束光滑优化问题，分析了两种投影梯度方案：一种采用固定步长，另一种采用回溯线搜索。固定步长方案在目标函数具有Lipschitz连续黎曼梯度的假设下进行分析，而回溯变体无需该假设即可证明聚点的平稳性。对于两种方案，我们都证明了在各自假设下，生成序列的每个聚点都是一阶平稳点，且无需可行集具备紧性；紧性仅用于确保聚点的存在性。当目标函数具有Lipschitz连续黎曼梯度时，我们推导了两种方案基于投影的平稳性度量的迭代复杂度界为\
英文摘要
We study constrained smooth optimization problems on Hadamard manifolds with closed geodesically convex feasible sets. We analyze two projected gradient schemes: one with a constant stepsize and another with a backtracking line search. The constant-stepsize scheme is analyzed under the assumption that the objective function has a Lipschitz continuous Riemannian gradient, whereas the backtracking variant does not require this assumption to establish stationarity of accumulation points. For both schemes, we prove that every accumulation point of the generated sequence is first-order stationary under the respective assumptions, without requiring compactness of the feasible set; compactness is needed only to ensure the existence of accumulation points. When the objective function has a Lipschitz continuous Riemannian gradient, we derive iteration-complexity bounds of order \(O(1/\sqrt{N})\) for projection-based stationarity measures for both schemes, together with the corresponding \(\varepsilon\)-complexity estimates. For the backtracking scheme, the complexity analysis additionally requires the trial line-search stepsizes to be uniformly bounded away from zero. Under the same respective assumptions, the generated sequences are also asymptotically regular. Finally, we illustrate the practical performance of the methods by solving constrained Karcher mean problems on the manifold of symmetric positive definite matrices.
Comments28 pages, 4 figures
