In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several scenarios.

翻译：本文针对紧致矩阵流形上的优化问题，提出了一种名为变换梯度投影（Transformed Gradient Projection, TGP）算法的新型算法框架，该算法利用投影到紧致矩阵流形上的操作实现。与现有算法相比，我们方法的核心创新在于采用一类新的搜索方向以及多种步长选择（包括Armijo步长、非单调Armijo步长和固定步长）来指导下一次迭代的选择。该框架具有灵活性，将经典梯度投影算法作为其特例，并与基于收缩的线搜索算法相交。特别地，我们重点关注Stiefel流形或Grassmann流形，揭示了文献中诸多现有算法均可视为我们提出框架中的具体实例，同时该算法框架还衍生出多个新的特例情形。随后，我们系统研究了这些算法在不同搜索方向和步长下的收敛性质。为此，我们深入分析了投影到紧致矩阵流形上的几何特性，从而将文献中与收缩相关的经典不等式拓展到新场景。基于这些发现，我们建立了TGP算法在三种不同步长下的弱收敛性、收敛速率和全局收敛性。当紧致矩阵流形为Stiefel流形或Grassmann流形时，我们的收敛结果涵盖或超越了现有文献中的结论。最后，通过一系列数值实验，我们观察到TGP算法由于在搜索方向选择上具有更大的灵活性，在多种场景下优于经典梯度投影算法和基于收缩的线搜索算法。