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

Fingerprint

Convex function
Monotone
Monotone Matrix
Open interval
Continuously differentiable
Convex Order
Second-order Cone
Optimization
Convex Cone
Monotonicity
Convexity
Cones
Verify
If and only if

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

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

In: Journal of Global Optimization, Vol. 45, No. 2, 01.10.2009, p. 259-279.

Research output: Contribution to journalArticle

Chen, Jein-Shan ; Chen, Xin ; Pan, Shaohua ; Zhang, Jiawei. / Some characterizations for SOC-monotone and SOC-convex functions. In: Journal of Global Optimization. 2009 ; Vol. 45, No. 2. pp. 259-279.
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