A Riemannian approach to lowrank algebraic Riccati equations
Authors:
B. Mishra and B. Vandereycken
Abstract
We propose a Riemannian optimization approach for computing lowrank solutions of the algebraic Riccati equation. The scheme alternates between fixedrank optimization and rankone updates. The fixedrank optimization is on the set of fixedrank 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 trustregion algorithm that is potentially scalable to largescale problems. The rankone update is based on a descent direction that ensures a monotonic decrease of the cost function. Preliminary numerical results on standard smallscale benchmarks show that we obtain solutions to the Riccati equation at lower ranks than the standard approaches.
Downloads
 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.
Note
We solve the equation $A^{T}X + XA + X BB^{T }X = C^{T}C$ for X on the lowrank 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.
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