Journal of Convex Analysis 32 (2025), No. 3, 937--950
Copyright Heldermann Verlag 2025
Metric Subregularity for a Piecewise Linear Mapping between two Banach Spaces
Xi Yin Zheng
Department of Mathematics, Yunnan University, Kunming, P. R. China
xyzheng@ynu.edu.cn
Yuhao Sun
Department of Mathematics, Yunnan University, Kunming, P. R. China
m15877941007_1@163.com
Wenqi Tang
Department of Mathematics, Yunnan University, Kunming, P. R. China
1309167448@qq.com
Observe that the graph of a piecewise linear mapping to an infinite dimensional Banach space is not the union of finitely many convex polyhedra. This and Robinson's theorem on local metric subregularity for a polyhedral mapping motivate us to consider the metric subregualrity for a piecewise linear mapping between two general Banach spaces. We prove that a piecewise linear mapping G between two Banach spaces is boundedly metrically subregular at any point in its graph gph(G) if and only if G is metrically subregular at some point in gph(G) if and only if G has the L-closed range property, which complements Robinson's theorem. As an complement of Mordukhovich's criterion on the metric regularity in the finite dimension case, we also provide a dual characterization for a piecewise linear mapping to be metrically regular.
Keywords: Piecewise linear mapping, metric subregularity, metric regularity.
MSC: 52B60, 52B70, 90C29.
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