A Riemannian approach to low-rank algebraic Riccati equations
B. Mishra and B. Vandereycken
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.
- Status: Extended abstract, proceedings of MTNS 2014.
- Paper: [Publisher’s pdf] [arXiv:1312.4883]
- Matlab code: Lowrank_algebraic_Riccati.zip
- Entry: Feb 24, 2014: The code is online.
We solve the equation $ATX + XA + X BBT X = CTC$ for X on the low-rank manifold of symmetric and positive definite matrices, where A is large and sparse, B is a tall matrix, and C is a fat matrix. For $B = 0$, this boils down to the continuous Lyapunov equation.
The code is built on the Manopt platform. Feel free to contact me on any issue.