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The aim of this paper is to derive convergence results for projectedline-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 Riemannianoptimization methods, which are successfully used on the smooth manifold$\mathcal{M}_k$ of rank-$k$ matrices, to its closure by taking steps alonggradient-related directions in the tangent cone, and afterwards projecting backto $\mathcal{M}_{\le k}$. Considering such a method circumvents thedifficulties which arise from the nonclosedness and the unbounded curvature of$\mathcal{M}_k$. The pointwise convergence is obtained for real-analyticfunctions on the basis of a \L{}ojasiewicz inequality for the projection of theantigradient to the tangent cone. If the derived limit point lies on the smoothpart of $\mathcal{M}_{\le k}$, i.e. in $\mathcal{M}_k$, this boils down to moreor less known results, but with the benefit that asymptotic convergence rateestimates (for specific step-sizes) can be obtained without an a prioricurvature 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 criticalpoints 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 $\nabla f(X) = 0$.

Convergence results for projected line-search methods on varieties of low-rank matrices via \L{}ojasiewicz inequality