Recent advancements in data science have significantly elevated the importance of orthogonally constrained optimization problems. The Riemannian approach has become a popular technique for addressing these problems due to the advantageous computational and analytical properties of the Stiefel manifold. Nonetheless, the interplay of nonsmoothness alongside orthogonality constraints introduces substantial challenges to current Riemannian methods, including scalability, parallelizability, complicated subproblems, and cumulative numerical errors that threaten feasibility. In this paper, we take a retraction-free primal-dual approach and propose a linearized smoothing augmented Lagrangian method specifically designed for nonsmooth and nonconvex optimization with orthogonality constraints. Our proposed method is single-loop and free of subproblem solving. We establish its iteration complexity of $O(ε^{-3})$ for finding $ε$-KKT points, matching the best-known results in the Riemannian optimization literature. Additionally, by invoking the standard Kurdyka-Lojasiewicz (KL) property, we demonstrate asymptotic sequential convergence of the proposed algorithm. Numerical experiments on both smooth and nonsmooth orthogonal constrained problems demonstrate the superior computational efficiency and scalability of the proposed method compared with state-of-the-art algorithms.

翻译：近年来数据科学的进展显著提升了正交约束优化问题的重要性。由于施蒂费尔流形具有有利的计算与分析性质，黎曼方法已成为处理此类问题的常用技术。然而，非光滑性与正交约束的相互作用给现有黎曼方法带来了重大挑战，包括可扩展性、可并行性、复杂子问题以及威胁可行性的累积数值误差。本文采用无回缩的原始-对偶策略，提出一种专为具有正交约束的非光滑非凸优化设计的线性化平滑增广拉格朗日方法。所提方法为单循环结构且无需求解子问题。我们建立了该方法寻找ε-KKT点的迭代复杂度为$O(ε^{-3})$，这与黎曼优化文献中已知的最优结果相匹配。此外，通过利用标准Kurdyka-Lojasiewicz（KL）性质，我们证明了所提算法的渐近序列收敛性。在光滑与非光滑正交约束问题上的数值实验表明，与最先进算法相比，所提方法具有优越的计算效率与可扩展性。