Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Xi Yin Zheng

Department of Mathematics, Yunnan University, Kunming, P. R. China

xyzheng@ynu.edu.cn

Wenqi Tang

Department of Mathematics, Yunnan University, Kunming, P. R. China

1309167448@qq.com

Keywords

  • Piecewise linear mapping
  • metric subregularity
  • metric regularity

Mathematics Subject Classification

52B60, 52B70, 90C29

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Suggested citation

X. Y. Zheng, Y. Sun, W. Tang. “Metric Subregularity for a Piecewise Linear Mapping between two Banach Spaces.” Journal of Convex Analysis 32 (2025), No. 3, 937–950. https://doi.org/10.68381/jca32049

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.