On Second-Order Cone Functions

We consider the second-order cone function (SOCF) $f:{\mathbb{R}}^n\to \mathbb{R}$ defined by $f(x)=c^{T}x+d-\mathrm{\parallel }Ax+b\mathrm{\parallel }$. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of $f$. The parameters $A\in {\mathbb{R}}^{m\times n}$ and $b\in {\mathbb{R}}^m$ are not uniquely determined. We show that every SOCF can be written in the form $f(x)=c^{T}x+d-\sqrt{{\delta }^2+(x-x_{\ast }{)}^{T}M(x-x_{\ast })}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $\delta$, $M=A^{T}A$, and $x_{\ast }$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above.