### 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 |
---|---|

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 |

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### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Global Optimization*,

*45*(2), 259-279. https://doi.org/10.1007/s10898-008-9373-z

**Some characterizations for SOC-monotone and SOC-convex functions.** / Chen, Jein-Shan; Chen, Xin; Pan, Shaohua; Zhang, Jiawei.

Research output: Contribution to journal › Article

*Journal of Global Optimization*, vol. 45, no. 2, pp. 259-279. https://doi.org/10.1007/s10898-008-9373-z

}

TY - JOUR

T1 - Some characterizations for SOC-monotone and SOC-convex functions

AU - Chen, Jein-Shan

AU - Chen, Xin

AU - Pan, Shaohua

AU - Zhang, Jiawei

PY - 2009/10/1

Y1 - 2009/10/1

N2 - 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.

AB - 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.

KW - SOC-convex function

KW - SOC-monotone function

KW - Second-order cone

UR - http://www.scopus.com/inward/record.url?scp=69949088315&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69949088315&partnerID=8YFLogxK

U2 - 10.1007/s10898-008-9373-z

DO - 10.1007/s10898-008-9373-z

M3 - Article

AN - SCOPUS:69949088315

VL - 45

SP - 259

EP - 279

JO - Journal of Global Optimization

JF - Journal of Global Optimization

SN - 0925-5001

IS - 2

ER -