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

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.