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

N1 - Copyright:
Copyright 2009 Elsevier B.V., All rights reserved.

PY - 2009/10

Y1 - 2009/10

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

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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 -