  The affine Grassmannian is a noncompact smooth manifold that parameterizes
all affine subspaces of a fixed dimension. It is a natural generalization of
Euclidean space, points being zero-dimensional affine subspaces. We will
realize the affine Grassmannian as a matrix manifold and extend Riemannian
optimization algorithms including steepest descent, Newton method, and
conjugate gradient, to real-valued functions on the affine Grassmannian. Like
their counterparts for the Grassmannian, these algorithms are in the style of
Edelman--Arias--Smith --- they rely only on standard numerical linear algebra
and are readily computable.
