# MathSciDoc: An Archive for Mathematician ∫

#### Optimization and Controlmathscidoc:1910.43876

Optimization, 55, (4), 363-385, 2006.8
Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function . We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., <i>f</i>(<i>t</i>) = <i>t</i> <sup>2</sup> <i>t</i> <sup>3</sup> 1/<i>t</i> <i>t</i> <sup>1/2</sup>, |<i>t</i>|, and [<i>t</i>]<sub>+</sub>. Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.
```@inproceedings{jein-shan2006the,
title={The convex and monotone functions associated with second-order cone},
author={Jein-Shan Chen},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020223254464943405},
booktitle={Optimization},
volume={55},
number={4},
pages={363-385},
year={2006},
}
```
Jein-Shan Chen. The convex and monotone functions associated with second-order cone. 2006. Vol. 55. In Optimization. pp.363-385. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020223254464943405.