  The recent interior point algorithm by Dahl and Andersen [10] for
nonsymmetric cones as well as earlier works [16,19] require derivative
information from the conjugate of the barrier function of the cones in the
problem. Besides a few special cases, there is no indication of when this
information is efficient to evaluate. We show how to compute the gradient of
the conjugate barrier function for seven useful nonsymmetric cones. In some
cases this is helpful for deriving closed-form expressions for the inverse
Hessian operator for the primal barrier.
