On Second-Order Cone Functions

TL;DR

This paper characterizes the strict concavity, boundedness, and parameter uniqueness of second-order cone functions, providing precise conditions and alternative representations for these functions in optimization contexts.

Contribution

It offers necessary and sufficient conditions for strict concavity, boundedness, and parameter identifiability of second-order cone functions, along with a canonical form representation.

Findings

Necessary and sufficient conditions for strict concavity.

Conditions for boundedness of SOCFs.

Parameter uniqueness criteria for SOCFs.

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.