论文标题
开放性，持有者指标规律性和持有人的连续性属性的固定图值
Openness, Holder metric regularity and Holder continuity properties of semialgebraic set-valued maps
论文作者
论文摘要
给定一个半gebraic set-valued map $ f \ colon \ colon \ mathbb {r}^n \ rightrightarrows \ mathbb {r}^m $带封闭图，我们表明地图$ f $是持有人在unders上，并且以下条件等于： （i）$ f $是从其域中进入其范围的开放地图，$ f $的范围是本地关闭的； （ii）地图$ f $是持有人的指数规则； （iii）逆映射$ f^{ - 1} $是伪持有人的连续； （iv）逆映射$ f^{ - 1} $是较低的伪持有器连续的。 通过鲁滨逊的正常地图公式，应用程序在半格式变异不等式的背景下导致以下结果：如果解决方案映射（作为参数矢量的映射）是较低的半连续性，则解决方案映射为有限且伪 - hölder连续。特别是，我们对Dontchev和Rockafellar \ Cite {Dontchev1996}的论文中提到的问题获得了负面答案。 作为副产品，我们表明，对于（不一定是半ge）连续的单值地图，从$ \ mathbb {r}^n $到$ \ mathbb {r}，$开放性和非肢体性是等效的。这一事实改善了Pühn\ cite {puhl1998}的主要结果，该{puhl1998}需要相关地图的凸度。
Given a semialgebraic set-valued map $F \colon \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ with closed graph, we show that the map $F$ is Holder metrically subregular and that the following conditions are equivalent: (i) $F$ is an open map from its domain into its range and the range of $F$ is locally closed; (ii) the map $F$ is Holder metrically regular; (iii) the inverse map $F^{-1}$ is pseudo-Holder continuous; (iv) the inverse map $F^{-1}$ is lower pseudo-Holder continuous. An application, via Robinson's normal map formulation, leads to the following result in the context of semialgebraic variational inequalities: if the solution map (as a map of the parameter vector) is lower semicontinuous then the solution map is finite and pseudo-Hölder continuous. In particular, we obtain a negative answer to a question mentioned in the paper of Dontchev and Rockafellar \cite{Dontchev1996}. As a byproduct, we show that for a (not necessarily semialgebraic) continuous single-valued map from $\mathbb{R}^n$ to $\mathbb{R},$ the openness and the non-extremality are equivalent. This fact improves the main result of Pühn \cite{Puhl1998}, which requires the convexity of the map in question.
