Paper ID sheet UCLINMA2021.06
 Title

On the continuity of the tangent cone to the determinantal variety
 Authors
 Guillaume Olikier, P.A. Absil
 Abstract

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 lowrank 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
 Lowrank matrices; Determinantal variety; Set convergence; Inner and outer limits; Setvalued mappings; Inner and outer semicontinuity; Tangent and normal cones
 Status
 SetValued and Variational Analysis, volume 30, pages 769–788 (2022)
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