On Second-Order Cone Functions

We consider the second-order cone function (SOCF) f : R n -> R defined by f x = c T x + d - A x + b , with parameters c is an element of R n , d is an element of R , A is an element of R m x n , and b is an element of R m . 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 f x = c T x + d - delta 2 + x - x & lowast; T M x - x & lowast; . We give necessary and sufficient conditions for the parameters c , d , delta , M = A T A , and x & lowast; to be uniquely determined. We also give necessary and sufficient conditions for f to be bounded above.