Optimization under the symplecticity constraint is an approach for solving various problems in quantum physics and scientific computing. Building on the results that this optimization problem can be transformed into an unconstrained problem on the symplectic Stiefel manifold, we construct geometric ingredients for Riemannian optimization with a new family of Riemannian metrics called tractable metrics and develop Riemannian Newton schemes. The newly obtained ingredients do not only generalize several existing results but also provide us with freedom to choose a suitable metric for each problem. To the best of our knowledge, this is the first try to develop the explicit second-order geometry and Newton's methods on the symplectic Stiefel manifold. For the Riemannian Newton method, we first consider novel operator-valued formulas for computing the Riemannian Hessian of a~cost function, which further allows the manifold to be endowed with a weighted Euclidean metric that can provide a preconditioning effect. We then solve the resulting Newton equation, as the central step of Newton's methods, directly via transforming it into a~saddle point problem followed by vectorization, or iteratively via applying any matrix-free iterative method either to the operator Newton equation or its saddle point formulation. Finally, we propose a hybrid Riemannian Newton optimization algorithm that enjoys both global convergence and quadratic/superlinear local convergence at the final stage. Various numerical experiments are presented to validate the proposed methods.

翻译：辛几何约束下的优化是解决量子物理和科学计算中各类问题的一种方法。基于该优化问题可转化为辛普莱斯蒂菲尔流形上无约束问题的已有结果，我们为黎曼优化构建了几何要素——采用一类称为可处理度量的新型黎曼度量，并发展了黎曼牛顿格式。新获得的要素不仅推广了若干现有结果，还为我们提供了针对具体问题选择合适度量的自由度。据我们所知，这是在辛普莱斯蒂菲尔流形上首次尝试建立显式二阶几何与牛顿方法。对于黎曼牛顿法，我们首先提出了计算代价函数黎曼海森矩阵的算子值公式新形式，该公式进一步允许流形配备具有预处理效果的加权欧氏度量。随后，我们通过两种途径求解作为牛顿方法核心步骤的牛顿方程：一是将其转化为鞍点问题后向量化直接求解，二是对算子牛顿方程或其鞍点形式应用任意无矩阵迭代法进行迭代求解。最后，我们提出了一种混合黎曼牛顿优化算法，该算法兼具全局收敛性与最终阶段的二次/超线性局部收敛性。通过多种数值实验验证了所提方法的有效性。