# A Riemannian approach to low-rank algebraic Riccati equations

arXiv: https://arxiv.org/abs/1312.4883 · HTML: https://mlsanity.com/paper/1312.4883/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2013.

## Built on by (most cited first, 1 of 1)

- [Low-Rank Updates of Matrix Square Roots](https://mlsanity.com/paper/2201.13156/) (2022)

## Abstract (arXiv)

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.