Abstract
In this paper, we study the parabolic second-order directional derivative in the Hadamard sense of a vector-valued function associated with circular cone. The vector-valued function comes from applying a given real-valued function to the spectral decomposition associated with circular cone. In particular, we present the exact formula of second-order tangent set of circular cone by using the parabolic second-order directional derivative of projection operator. In addition, we also deal with the relationship of second-order differentiability between the vector-valued function and the given real-valued function. The results in this paper build fundamental bricks to the characterizations of second-order necessary and sufficient conditions for circular cone optimization problems.
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Acknowledgements
The authors are gratefully indebted to the anonymous referee for their valuable suggestions and remarks that allowed us to improve the original presentation of the paper. The first author’s work is supported by National Natural Science Foundation of China (11101248, 11271233), Shandong Province Natural Science Foundation (ZR2016AM07), and Young Teacher Support Program of Shandong University of Technology. The second author’s work is supported by Basic and Frontier Technology Research Project of Henan Province (162300410071). The third author’s work is supported by Ministry of Science and Technology, Taiwan.
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Communicated by Byung-Soo Lee.
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Zhou, J., Tang, J. & Chen, JS. Parabolic Second-Order Directional Differentiability in the Hadamard Sense of the Vector-Valued Functions Associated with Circular Cones. J Optim Theory Appl 172, 802–823 (2017). https://doi.org/10.1007/s10957-016-0935-9
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DOI: https://doi.org/10.1007/s10957-016-0935-9