  Conditional graph entropy is known to be the minimal rate for a natural
functional compression problem with side information at the receiver. In this
paper we show that it can be formulated as an alternating minimization problem,
which gives rise to a simple iterative algorithm for numerically computing
(conditional) graph entropy. This also leads to a new formula which shows that
conditional graph entropy is part of a more general framework: the solution of
an optimization problem over a convex corner. In the special case of graph
entropy (i.e., unconditioned version) this was known due to Csisz\'ar,
K\"orner, Lov\'asz, Marton, and Simonyi. In that case the role of the convex
corner was played by the so-called vertex packing polytope. In the conditional
version it is a more intricate convex body but the function to minimize is the
same. Furthermore, we describe a dual problem that leads to an optimality check
and an error bound for the iterative algorithm.
