Some characterizations for SOC-monotone and SOC-convex functions

Jein-Shan Chen, Xin Chen, Shaohua Pan, Jiawei Zhang

Research output: Contribution to journalArticle

15 Citations (Scopus)

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 languageEnglish
Pages (from-to)259-279
Number of pages21
JournalJournal of Global Optimization
Volume45
Issue number2
DOIs
Publication statusPublished - 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

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