On Second-Order Cone Functions (2308.01360v2)

Abstract: We consider the second-order cone function (SOCF) $f: {\mathbb R}^n \to \mathbb R$ defined by $f(x)= c^T x + d -|A x + b |$. 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 $f(x) = c^T x + d -\sqrt{\delta^2 + (x-x_)^TM(x-x_)}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $\delta$, $M = A^T A$, and $x_*$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above.