Abstract
Let \(\mathcal{L}_{\theta }\) be the circular cone in \({\mathbb {R}}^n\) which includes second-order cone as a special case. For any function f from \({\mathbb {R}}\) to \({\mathbb {R}}\), one can define a corresponding vector-valued function \(f^{\mathcal{L}_{\theta }}\) on \({\mathbb {R}}^n\) by applying f to the spectral values of the spectral decomposition of \(x \in {\mathbb {R}}^n\) with respect to \(\mathcal{L}_{\theta }\). The main results of this paper are regarding the H-differentiability and calmness of circular cone function \(f^{\mathcal{L}_{\theta }}\). Specifically, we investigate the relations of H-differentiability and calmness between f and \(f^{\mathcal{L}_{\theta }}\). In addition, we propose a merit function approach for solving the circular cone complementarity problems under H-differentiability. These results are crucial to subsequent study regarding various analysis towards optimizations associated with circular cone.
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Acknowledgments
We are gratefully indebted to anonymous referees and the editor for their valuable suggestions that help us to essentially improve the paper a lot. In particular, the content of Sect. 5 is inspired by one referee’s suggestion.
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Jinchuan Zhou’s work is supported by National Natural Science Foundation of China (11101248, 11271233, 11171247) and Shandong Province Natural Science Foundation (ZR2012AM016).
Jein-Shan Chen’s work is supported by Ministry of Science and Technology, Taiwan.
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Zhou, J., Chang, YL. & Chen, JS. The H-differentiability and calmness of circular cone functions. J Glob Optim 63, 811–833 (2015). https://doi.org/10.1007/s10898-015-0312-5
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DOI: https://doi.org/10.1007/s10898-015-0312-5