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 $f_t:=f-tg$ 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.

中文翻译：

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非孤立奇点的莫尔斯理论方法和优化应用

设X是一个复杂的仿射变体${\mathbb{C}}^N$， 然后让 $F：{\mathbb{C}}^N\to \mathbb{C}$是一个多项式函数，其对X的限制是非常量的。为了$G：{\mathbb{C}}^N\to \mathbb{C}$ 一个一般的线性函数，我们研究了单参数族的临界点的极限行为 $F_{吨}：=F-吨G$ 作为 $吨\to 0$. 我们的主要结果根据g对奇异地层的限制的临界集给出了这个限制的表达$(X,F)$. 我们将此结果应用于优化问题的上下文中。例如，我们考虑仿射变体和可能的非泛型数据点的最近点问题（例如，欧几里得距离度）。