This article is concerned with the problem of minimizing a smooth function over the Stiefel manifold. In order to address this problem, we introduce two adaptive scaled gradient projection methods that incorporate scaling matrices that depend on the step-size and a parameter that controls the search direction. These iterative algorithms use a projection operator based on the QR factorization to preserve the feasibility in each iteration. However, for some particular cases, the proposals do not require the use of any projection operator. In addition, we consider a Barzilai and Borwein-like step-size combined with the Zhang–Hager nonmonotone line-search technique in order to accelerate the convergence of the proposed procedures. We proved the global convergence for these schemes, and we evaluate their effectiveness and efficiency through an extensive computational study, comparing our approaches with other state-of-the-art gradient-type algorithms.

本文涉及使Stiefel流形上的平滑函数最小化的问题。为了解决这个问题，我们介绍了两种自适应缩放梯度投影方法，这些方法结合了取决于步长和控制搜索方向的参数的缩放矩阵。这些迭代算法使用基于QR分解的投影算子来保留每次迭代中的可行性。但是，对于某些特定情况，建议不需要使用任何投影运算符。另外，我们考虑将Barzilai和Borwein样的步长与Zhang-Hager非单调线搜索技术相结合，以加快所提出程序的收敛性。我们证明了这些方案的全球融合，