In this paper, we focus on building an optimization scheme over the Stiefel manifold that maintains each iterate feasible. We focus on conjugate gradient methods and compare our scheme to the Riemannian optimization approach. We parametrize the Stiefel manifold using the polar decomposition to build an optimization problem over a vector space, instead of a Riemannian manifold. The result is a conjugate gradient method that averts the use of a vector transport, needed in the Riemannian conjugate gradient method. The performance of our method is tested on a variety of numerical experiments and compared with those of three Riemannian optimization methods.

在本文中，我们着重于在Stiefel流形上构建优化方案，以保持每个迭代的可行性。我们专注于共轭梯度法，并将我们的方案与黎曼优化方法进行比较。我们使用极性分解对Stiefel流形进行参数化，以建立向量空间而不是黎曼流形的优化问题。结果是共轭梯度方法，该方法避免了使用黎曼共轭梯度方法所需的矢量传输。我们的方法的性能在各种数值实验中进行了测试，并与三种黎曼优化方法的性能进行了比较。