关于行列式簇的切锥连续性的研究
On the continuity of the tangent cone to the determinantal variety
AI总结：
本文通过计算内外极限研究秩至多为r的实矩阵集合上切锥与法锥对应关系的连续性，并将其与实行列式簇的Whitney分层a-正则性联系起来。
摘要：
切锥和法锥在约束优化中起着重要作用，用于描述可行搜索方向，特别是用于表述最优性条件。它们显著出现在近年来针对光滑与非光滑低秩优化的多种算法中，其中可行集为所有秩至多为 $r$ 的 $m \times n$ 实矩阵集合 $\mathbb{R}_{\leq r}^{m \times n}$。在本文中，受此类算法收敛性分析的启发，我们通过计算内极限和外极限，研究了将每个 $X \in \mathbb{R}_{\leq r}^{m \times n}$ 映射到 $\mathbb{R}_{\leq r}^{m \times n}$ 在 $X$ 处切锥的对应关系的连续性。我们还推导出相应法锥对应关系的连续性结果。最后，我们证明我们的结果将 $\mathbb{R}_{\leq r}^{m \times n}$ 的 Whitney 分层的 $a$-正则性作为一个特例包含在内，这源于该集合是一个实代数簇，称为实行列式簇。
英文摘要：
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.
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