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：黎曼优化关注的是自变量位于光滑流形上的问题。数值线性代数中有许多问题属于此类，其中流形通常由特殊的矩阵结构（如正交性或正定性）所定义。沿着这一研究方向，我们研究了辛斯蒂菲尔流形上的黎曼优化工具。我们针对辛斯蒂菲尔流形设计了一种黎曼信赖域方法，以补充现有的数值优化算法集合。为此，我们在右不变度量下推导了黎曼海森矩阵的矩阵表达式。此外，我们提出了一种新的收缩映射来逼近黎曼测地线。最后，我们开展了一项比较研究，在具有辛约束的选定矩阵优化问题上，对比了最速下降法、共轭梯度法和信赖域法的黎曼变体性能。