Some characterizations for SOC-monotone and SOC-convex functions |
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Authors: | Jein-Shan Chen Xin Chen Shaohua Pan Jiawei Zhang |
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Affiliation: | 1.Department of Mathematics,National Taiwan Normal University,Taipei,Taiwan;2.Department of Industrial and Enterprise System Engineering,University of Illinois at Urbana–Champaign,Urbana,USA;3.School of Mathematical Sciences,South China University of Technology,Guangzhou,China;4.Department of Information, Operations and Management Sciences,New York University,New York,USA |
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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 ≥ 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. |
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