Some characterizations for SOC-monotone and SOC-convex functions

Jein-Shan Chen Xin Chen Shaohua Pan Jiawei Zhang

Spectral Theory and Operator Algebra mathscidoc:1910.43892

Journal of Global Optimization, 45, (2), 259-279, 2009.10
We provide some characterizations for SOC-monotone and SOC-convex functions by using differential analysis. From these characterizations, we particularly obtain that a continuously differentiable function defined in an open interval is SOC-monotone (SOC-convex) of order <i>n</i> 3 if and only if it is 2-matrix monotone (matrix convex), and furthermore, such a function is also SOC-monotone (SOC-convex) of order <i>n</i> 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363385, 2006) does not hold in general. Some examples are included to illustrate that these characterizations open convenient ways to verify the SOC-monotonicity and the SOC-convexity of a continuously differentiable function defined on an open interval, which are often involved in the solution methods of the convex second-order cone optimization.
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@inproceedings{jein-shan2009some,
  title={Some characterizations for SOC-monotone and SOC-convex functions},
  author={Jein-Shan Chen, Xin Chen, Shaohua Pan, and Jiawei Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020223908361029421},
  booktitle={Journal of Global Optimization},
  volume={45},
  number={2},
  pages={259-279},
  year={2009},
}
Jein-Shan Chen, Xin Chen, Shaohua Pan, and Jiawei Zhang. Some characterizations for SOC-monotone and SOC-convex functions. 2009. Vol. 45. In Journal of Global Optimization. pp.259-279. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020223908361029421.
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