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Openness, Holder metric regularity and Holder continuity properties of semialgebraic set-valued maps

Jae Hyoung Lee (mc7558***at***naver.com)
Tien-Son Pham (sonpt***at***dlu.edu.vn)

Abstract: 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 Holder continuous; (iv) the inverse map $F^{-1}$ is lower 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-H\"older 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 P\"uhn \cite{Puhl1998}, which requires the convexity of the map in question.

Keywords: openness, Holder metric (sub)regularity, Holder continuity, variational inequality, semialgebraicity

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Category 2: Complementarity and Variational Inequalities


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Entry Submitted: 04/04/2020
Entry Accepted: 04/04/2020
Entry Last Modified: 04/14/2020

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