- Openness, Holder metric regularity and Holder continuity properties of semialgebraic set-valued maps Jae Hyoung Lee (mc7558naver.com) Tien-Son Pham (sonptdlu.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 Citation: Download: [PDF]Entry Submitted: 04/04/2020Entry Accepted: 04/04/2020Entry Last Modified: 04/14/2020Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.