On the continuity of the tangent cone to the determinantal variety
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910584731074560 |
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| author | Olikier, Guillaume Absil, P. -A. |
| author_facet | Olikier, Guillaume Absil, P. -A. |
| contents | 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 $\mathbb{R}_{\leq r}^{m \times n}$ 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 $X \in \mathbb{R}_{\leq r}^{m \times n}$ to the tangent cone to $\mathbb{R}_{\leq r}^{m \times n}$ 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 $\mathbb{R}_{\leq r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_03979 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the continuity of the tangent cone to the determinantal variety Olikier, Guillaume Absil, P. -A. Optimization and Control Numerical Analysis 14M12, 15B99, 26E25, 49J53 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 $\mathbb{R}_{\leq r}^{m \times n}$ 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 $X \in \mathbb{R}_{\leq r}^{m \times n}$ to the tangent cone to $\mathbb{R}_{\leq r}^{m \times n}$ 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 $\mathbb{R}_{\leq r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety. |
| title | On the continuity of the tangent cone to the determinantal variety |
| topic | Optimization and Control Numerical Analysis 14M12, 15B99, 26E25, 49J53 |
| url | https://arxiv.org/abs/2201.03979 |