论文标题
一种摩尔斯理论方法，用于非分离奇异性和优化应用
A Morse theoretic approach to non-isolated singularities and applications to optimization
论文作者
论文摘要
令$ x $为$ \ mathbb {c}^n $中的复杂仿射品种，让$ f：\ mathbb {c}^n \ to \ mathbb {c} $是一个多项式函数，其限制对$ x $是非consant的。对于$ g：\ mathbb {c}^n \ to \ mathbb {c} $一般线性函数，我们研究了$ f_t的一个参数家族的关键点的限制行为：= f_t $ = f-tg $ as $ t \ as $ t \至0 $。我们的主要结果就$ g $ $ g $的关键集对$（x，f）$的关键集进行了表达。我们将此结果应用于优化问题的背景。例如，我们考虑仿射品种和可能是非元素数据点的最近点问题（例如欧几里得距离度）。
Let $X$ be a complex affine variety in $\mathbb{C}^N$, and let $f:\mathbb{C}^N\to \mathbb{C}$ be a polynomial function whose restriction to $X$ is nonconstant. For $g:\mathbb{C}^N \to \mathbb{C}$ a general linear function, we study the limiting behavior of the critical points of the one-parameter family of $f_t: =f-tg$ as $t\to 0$. Our main result gives an expression of this limit in terms of critical sets of the restrictions of $g$ to the singular strata of $(X,f)$. We apply this result in the context of optimization problems. For example, we consider nearest point problems (e.g., Euclidean distance degrees) for affine varieties and a possibly nongeneric data point.
