论文标题
关于切线锥与确定性品种的连续性
On the continuity of the tangent cone to the determinantal variety
论文作者
论文摘要
切线和正常锥体在约束优化中起着重要作用，以描述可接受的搜索方向，尤其是制定最佳条件。它们显然出现在各种最近的算法中，用于平滑和非平滑低级别优化，其中可行的集合是$ \ mathbb {r} _ {\ leq r}^{m \ times n} $的所有$ m \ times n $ times n $ n $ Real n $ Real n $ Real n $ r $ r $ r $ $ $。在本文中，通过对这种算法的收敛分析的激励，我们通过计算内部和外部限制，将每个$ x \ in \ Mathbb {r} _ {\ leq r} _ {\ leq r}^{m \ times n} $映射到$ \ \ time n}到$ \ mathbb {mathbb {r} r} r的连续性。 n} $ at $ x $。我们还推断出有关相应正常锥对应关系的连续性的结果。最后，我们表明我们的结果包括$ \ Mathbb {r} _ {\ leq r}^{m \ times n} $的惠特尼分层的$ a $ regormuntiation的$ a $ regormuntialition，这是从此集合中是一个真正的代数品种，称为真实的确定性品种。
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.
