  The aim of this paper is to derive convergence results for projected
line-search methods on the real-algebraic variety $\mathcal{M}_{\le k}$ of real
$m \times n$ matrices of rank at most $k$. Such methods extend Riemannian
optimization methods, which are successfully used on the smooth manifold
$\mathcal{M}_k$ of rank-$k$ matrices, to its closure by taking steps along
gradient-related directions in the tangent cone, and afterwards projecting back
to $\mathcal{M}_{\le k}$. Considering such a method circumvents the
difficulties which arise from the nonclosedness and the unbounded curvature of
$\mathcal{M}_k$. The pointwise convergence is obtained for real-analytic
functions on the basis of a \L{}ojasiewicz inequality for the projection of the
antigradient to the tangent cone. If the derived limit point lies on the smooth
part of $\mathcal{M}_{\le k}$, i.e. in $\mathcal{M}_k$, this boils down to more
or less known results, but with the benefit that asymptotic convergence rate
estimates (for specific step-sizes) can be obtained without an a priori
curvature bound, simply from the fact that the limit lies on a smooth manifold.
At the same time, one can give a convincing justification for assuming critical
points to lie in $\mathcal{M}_k$: if $X$ is a critical point of $f$ on
$\mathcal{M}_{\le k}$, then either $X$ has rank $k$, or $\nabla f(X) = 0$.
