We study the convex duality method for robust utility maximization in the
presence of a random endowment. When the underlying price process is a locally
bounded semimartingale, we show that the fundamental duality relation holds
true for a wide class of utility functions on the whole real line and unbounded
random endowment. To obtain this duality, we prove a robust version of
Rockafellar's theorem on convex integral functionals and apply Fenchel's
general duality theorem.