A Riemannian approach to low-rank algebraic Riccati equations
A Riemannian approach to low-rank algebraic Riccati equations
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低阶代数 Riccati 方程的黎曼方法
DOI：
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发表时间：
2013
期刊：
影响因子：
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通讯作者：
Bart Vandereycken
中科院分区：
文献类型：
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作者：
Bamdev Mishra;Bart Vandereycken
提出了一种计算代数Riccati方程低阶解的黎曼最优化方法。该方案在固定秩优化和秩1更新之间交替。固定秩优化是关于固定秩对称正定矩阵的集合，该集合被赋予特定的黎曼度量(和几何)，该度量(和几何)与目标函数的结构相适应。我们具体讨论了黎曼信赖域算法的实现，该算法具有潜在的可伸缩到大规模问题的能力。排名1的更新基于确保成本函数单调减小的下降方向。在标准小尺度基准上的初步数值结果表明，我们以比标准方法更低的阶数获得了Riccati方程的解。
We propose a Riemannian optimization approach for computing low-rank solutions of the algebraic Riccati equation. The scheme alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is on the set of fixed-rank symmetric positive definite matrices which is endowed with a particular Riemannian metric (and geometry) that is tuned to the structure of the objective function. We specifically discuss the implementation of a Riemannian trust-region algorithm that is potentially scalable to large-scale problems. The rank-one update is based on a descent direction that ensures a monotonic decrease of the cost function. Preliminary numerical results on standard small-scale benchmarks show that we obtain solutions to the Riccati equation at lower ranks than the standard approaches.