Openness, Hölder Metric Regularity, and Hölder Continuity Properties of Semialgebraic Set-Valued Maps
Jae‐Hyoung Lee, Tiến-Sơn Phạm · SIAM Journal on Optimization · 2022
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 Hölder 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 Hölder metrically regular; (iii) the inverse map $F^{-1}$ is pseudo-Hölder continuous; (iv) the inverse map $F^{-1}$ 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-Hölder continuous. In particular, we obtain a negative answer to a question mentioned in the paper of Dontchev and Rockafellar SIAM J. Optim., 4 (1996), pp. 1087--1105. 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 nonextremality are equivalent. This fact improves the main result of Pühl J. Math. Anal. Appl., 227 (1998), pp. 382--395, which requires the convexity of the map in question.