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

Abstract: Given a semialgebraic set-valued map with closed graph, we show that it is Hölder metrically subregular and that the following conditions are equivalent:

(i) the map is an open map from its domain into its range and the range of is locally closed;

(ii) the map is Hölder metrically regular;

(iii) the inverse map is pseudo-Hölder continuous;

(iv) the inverse map is lower pseudo-Hölder 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-Holder continuous. In particular, we obtain a negative answer to a question mentioned in the paper of Dontchev and Rockafellar [Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim., 4(4):1087–1105, 1996]. As a byproduct, we show that for a (not necessarily semialgebraic) continuous single-valued map, the openness and the non-extremality are equivalent. This fact improves the main result of Pühn [Convexity and openness with linear rate. J. Math. Anal. Appl., 227:382–395, 1998], which requires the convexity of the map in question.

optimization and control

Audience: researchers in the topic

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