Some characterizations for SOC-monotone and SOC-convex functions

  • Authors:

  • Jein-Shan Chen;Xin Chen;Shaohua Pan;Jiawei Zhang

  • Affiliations:

  • Department of Mathematics, National Taiwan Normal University, Taipei, Taiwan 11677;Department of Industrial and Enterprise System Engineering, University of Illinois at Urbana---Champaign, Urbana, USA 61801;School of Mathematical Sciences, South China University of Technology, Guangzhou, China 510640;Department of Information, Operations and Management Sciences, New York University, New York, USA 10012-1126

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2009

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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 驴 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 驴 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.