  The Hessian of a differentiable convex function is positive semidefinite.
Therefore, checking the Hessian of a given function is a natural approach to
certify convexity. However, implementing this approach is not straightforward
since it requires a representation of the Hessian that allows its analysis.
Here, we implement this approach for a class of functions that is rich enough
to support classical machine learning. For this class of functions, it was
recently shown how to compute computational graphs of their Hessians. We show
how to check these graphs for positive semidefiniteness. We compare our
implementation of the Hessian approach with the well-established disciplined
convex programming (DCP) approach and prove that the Hessian approach is at
least as powerful as the DCP approach for differentiable functions.
Furthermore, we show for a state-of-the-art implementation of the DCP approach
that, for differentiable functions, the Hessian approach is actually more
powerful. That is, it can certify the convexity of a larger class of
differentiable functions.
