On Second-Order Cone Functions

Abstract

We consider the second-order cone function (SOCF) f:Rn⟶R defined by fx=cTx+d−Ax+b, with parameters c∈Rn, d∈R, A∈Rm×n, and b∈Rm. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of f. The parameters A and b are not uniquely determined. We show that every SOCF can be written in the form fx=cTx+d−δ2+x−x∗TMx−x∗. We give necessary and sufficient conditions for the parameters c, d, δ, M=ATA, and x∗ to be uniquely determined. We also give necessary and sufficient conditions for f to be bounded above.