Paper ID sheet UCL-INMA-2021.06


On the continuity of the tangent cone to the determinantal variety

Guillaume Olikier, P.-A. Absil
Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set $\mathbb{R}_{\le r}^{m \times n}$ of all $m \times n$ real matrices of rank at most $r$. In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each $X \in \mathbb{R}_{\le r}^{m \times n}$ to the tangent cone to $\mathbb{R}_{\le r}^{m \times n}$ at $X$. We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the $a$-regularity of the Whitney stratification of $\mathbb{R}_{\le r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety.
Key words
Low-rank matrices; Determinantal variety; Set convergence; Inner and outer limits; Set-valued mappings; Inner and outer semicontinuity; Tangent and normal cones
Set-Valued and Variational Analysis, volume 30, pages 769–788 (2022)