论文标题
对符号齿状歧管的优化：基于SR分解的回缩和应用
Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications
论文作者
论文摘要
在光学，量子物理学，稳定性分析和动力学系统控制方面的许多问题可以带到矩阵变量受到符号结构约束的优化问题。由于此约束很好地形成了所谓的符号符号齿状歧管，因此优先优化了Riemannian优化，因为在准备必要的几何工具后，可以从无约束的优化方法中借用想法。可以说，缩回是决定迭代方式更新给定搜索方向的最重要的。到目前为止，已经构建了两次缩回：一个依赖于Cayley变换，另一种是使用准地球曲线设计的。在本文中，我们提出了一种基于SR矩阵分解的新缩回。我们证明其域包含开放的单元球，这对于证明基于梯度的优化算法的全局收敛至关重要。此外，我们考虑了三种应用：惊直靶矩阵问题，符合性特征值计算和哈密顿系统的符号模型降低 - 以及各种示例。广泛的数值比较揭示了所提出的优化算法的强度。
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.
