Numerous problems in optics, quantum physics, stability analysis, and control of dynamical systems can be brought to an optimization problem with matrix variable subjected to the symplecticity constraint. As this constraint nicely forms a so-called symplectic Stiefel manifold, Riemannian optimization is preferred, because one can borrow ideas from unconstrained optimization methods after preparing necessary geometric tools. Retraction is arguably the most important one which decides the way iterates are updated given a search direction. Two retractions have been constructed so far: one relies on the Cayley transform and the other is designed using quasi-geodesic curves. In this paper, we propose a new retraction which is based on an SR matrix decomposition. We prove that its domain contains the open unit ball which is essential in proving the global convergence of the associated gradient-based optimization algorithm. Moreover, we consider three applications--symplectic target matrix problem, symplectic eigenvalue computation, and symplectic model reduction of Hamiltonian systems--with various examples. The extensive numerical comparisons reveal the strengths of the proposed optimization algorithm.
翻译:光学、量子物理学、稳定性分析和动态系统控制方面的许多问题都有可能变成最优化问题,因为矩阵变异会受干扰制约。由于这一限制很恰当地形成了所谓的间歇性Stiefel 元体,因此里曼尼亚优化是首选的,因为人们可以在准备必要的几何工具后从不受限制的优化方法中借出想法。回缩可以说是决定以搜索方向更新迭代方式的最重要问题。迄今为止,已经建造了两套撤回程序:一个依靠Cayley变形,另一个使用准地球曲线来设计。我们在本文件中提出一个新的回缩,以SR矩阵变形为基础。我们证明,其域包含开放的单位球,这对于证明相关的梯度优化算法的全球趋同至关重要。此外,我们考虑了三种应用-中位性目标矩阵问题,即静脉冲电子值计算,以及汉密尔顿系统模拟变形模型的缩减,并有多种实例。广泛的数字比较显示了拟议的压法的强度。