Metric subregularity and/or calmness of the normal cone mapping to the p-order conic constraint system

  • Ying Sun
  • Shaohua Pan
  • Shujun BiEmail author
Original Paper


For a finite convex function, we show that its subdifferential mapping is metrically subregular if and only if the normal cone mapping to its epigraph is metrically subregular. Then, for the nonconvex composition \(\psi =\theta \circ G\) where G is a continuously differentiable mapping and \(\theta \) is an extended real-valued function, we develop a criterion to identify the metric subregularity and calmness of the subdifferential mapping of \(\psi \) in terms of that of the subdifferential mapping of \(\theta \). Together with the existing results, we obtain the metric subregularity of the normal cone mapping to the vector and matrix p-order cone \(K_p\) and the conic constraint system \(g^{-1}(K_p)\) with \(p\in [1,2]\cup \{+\infty \}\), where g is a continuously differentiable mapping. We also establish the calmness of the normal cone mapping to \(K_p\) and \(g^{-1}(K_p)\) with \(p\in [2,+\infty ]\cup \{1\}\).


Normal cone mapping p-Order cone constraint systems Metric subregularity Calmness 



The authors would like to express their sincere thanks to anonymous referees for valuable suggestions and comments for the paper. The research of S. H. Pan was supported by the National Natural Science Foundation of China under Project No. 11571120 and the Natural Science Foundation of Guangdong Province under Project No. 2015A030313214, and S. J. Bi was supported by the National Natural Science Foundation of China under Project Nos. 11701186 and 11561015, and the Natural Science Foundation of Guangdong Province under Project No. 2017A030310418, and the Natural Science Foundation of Guangxi Province under Project No. 2016GXNSFFA380009.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.South China University of TechnologyGuangzhouChina

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