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

Given a semialgebraic set-valued map F colon ℝn rightrightarrows ℝm 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-H"older continuous. In particular, we obtain a negative answer to a question mentioned in the paper of Dontchev and Rockafellar citeDontchev1996. As a byproduct, we show that for a (not necessarily semialgebraic) continuous single-valued map from ℝn to ℝ, the openness and the non-extremality are equivalent. This fact improves the main result of P"uhn citePuhl1998, which requires the convexity of the map in question.