On the Continuity of the Tangent Cone to the Determinantal Variety

• Published in: Set-valued and variational analysis Vol. 30; no. 2; pp. 769 - 788
• Main Authors: Olikier, Guillaume, Absil, P.-A.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-06-2022; Springer Nature B.V
• Subjects: Algorithms; Analysis; Cones; Continuity (mathematics); Mathematics; Mathematics and Statistics; Optimization; Original Research; 14M12; Determinantal variety; 49J53; 15B99; Inner and outer limits; Tangent and normal cones; Set-valued mappings; Low-rank matrices; 26E25; Set convergence; Inner and outer semicontinuity
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-021-00618-9

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
ℝ
≤
r
m
×
n
of all
m
×
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
∈
ℝ
≤
r
m
×
n
to the tangent cone to
ℝ
≤
r
m
×
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
ℝ
≤
r
m
×
n
following from the fact that this set is a real algebraic variety, called the real determinantal variety.