  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.
