## Abstract

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 n ≥ 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 n ≤ 2 if it is 2-matrix monotone (matrix convex). In addition, we also prove that Conjecture 4.2 proposed in Chen (Optimization 55:363-385, 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.

Original language | English |
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Pages (from-to) | 259-279 |

Number of pages | 21 |

Journal | Journal of Global Optimization |

Volume | 45 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2009 Oct 1 |

## Keywords

- SOC-convex function
- SOC-monotone function
- Second-order cone

## ASJC Scopus subject areas

- Computer Science Applications
- Management Science and Operations Research
- Control and Optimization
- Applied Mathematics