Riemannian optimization is concerned with problems, where the independent variable lies on a smooth manifold. There is a number of problems from numerical linear algebra that fall into this category, where the manifold is usually specified by special matrix structures, such as orthogonality or definiteness. Following this line of research, we investigate tools for Riemannian optimization on the symplectic Stiefel manifold. We complement the existing set of numerical optimization algorithms with a Riemannian trust region method tailored to the symplectic Stiefel manifold. To this end, we derive a matrix formula for the Riemannian Hessian under a right-invariant metric. Moreover, we propose a novel retraction for approximating the Riemannian geodesics. Finally, we conduct a comparative study in which we juxtapose the performance of the Riemannian variants of the steepest descent, conjugate gradients, and trust region methods on selected matrix optimization problems that feature symplectic constraints.

翻译：黎曼优化研究的是自变量位于光滑流形上的优化问题。数值线性代数中的许多问题都属于这一范畴，其流形通常由特殊的矩阵结构（如正交性或正定性）定义。沿着这一研究方向，我们探究了用于辛Stiefel流形的黎曼优化工具。我们通过一种针对辛Stiefel流形定制的黎曼信赖域方法，对现有的数值优化算法进行了补充。为此，我们在右不变度量下推导了黎曼海森矩阵的矩阵公式。此外，我们提出了一种新颖的收缩方法来近似黎曼测地线。最后，我们开展了一项对比研究，将黎曼最速下降法、共轭梯度法和信赖域法在涉及辛约束的选定的矩阵优化问题上的性能进行了并列比较。