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，提出了一种称为变换梯度投影（TGP）算法的新型算法框架，该框架利用到该紧致矩阵流形的投影。与现有算法相比，我们方法的关键创新在于利用了一类新的搜索方向以及多种步长（包括Armijo步长、非单调Armijo步长和固定步长）来指导下一次迭代点的选取。我们的框架具有灵活性，它将经典的梯度投影算法作为特例包含在内，并与基于回撤的线搜索算法存在交集。值得注意的是，我们重点关注Stiefel流形或Grassmann流形，揭示了文献中许多现有算法可视为我们提出框架内的特定实例，并且该算法框架也引出了几个新的特例。接着，我们深入探讨了这些算法在不同搜索方向和步长下的收敛性质。为此，我们广泛分析了到紧致矩阵流形的投影的几何性质，从而能够推广文献中与回撤相关的经典不等式。基于这些见解，我们在三种不同步长下建立了TGP算法的弱收敛性、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，我们的收敛结果要么涵盖、要么超越了文献中的结果。最后，通过一系列数值实验，我们观察到，由于TGP算法在选择搜索方向上具有更高的灵活性，在多种场景下其性能优于经典的梯度投影算法和基于回撤的线搜索算法。