2020/04/05 by Jae‐Hyoung Lee, Lee, Jae Hyoung, Tiến-Sơn Phạm +1 · 1 citation
Computer Science · Mathematics · #14P10 #49J40 #49J52 #49J53 #90C31 #Analytic and geometric function theory #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2004.02188
openalex publication_date 2020/04/05 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
Given a semialgebraic set-valued map F colon \ℝn rightrightarrows\n\ℝm with closed graph, we show that the map F is Holder metrically\nsubregular and that the following conditions are equivalent:\n (i) F is an open map from its domain into its range and the range of F is\nlocally closed;\n (ii) the map F is Holder metrically regular;\n (iii) the inverse map F-1 is pseudo-Holder continuous;\n (iv) the inverse map F-1 is lower pseudo-Holder continuous.\n An application, via Robinson's normal map formulation, leads to the following\nresult in the context of semialgebraic variational inequalities: if the\nsolution map (as a map of the parameter vector) is lower semicontinuous then\nthe solution map is finite and pseudo-H "older continuous. In particular, we\nobtain a negative answer to a question mentioned in the paper of Dontchev and\nRockafellar citeDontchev1996.\n As a byproduct, we show that for a (not necessarily semialgebraic) continuous\nsingle-valued map from \ℝn to \ℝ, the openness and the\nnon-extremality are equivalent. This fact improves the main result of P "uhn\n citePuhl1998, which requires the convexity of the map in question.\n