This chapter explores entropy optimization within the framework of convex analysis and partially finite convex programming. It focuses on entropy-like functionals under moment constraints, employing function spaces defined via integrability conditions. A central result is an extension of Fenchel duality, which enables the resolution of entropy optimization problems in infinite-dimensional settings with finite constraints. The chapter systematically develops convex conjugacy and duality principles, particularly in the context of integral functionals, drawing from Rockafellar’s work. Applications include the Maximum Entropy on the Mean method, which illustrates the practical impact of these theoretical developments. The discussion bridges convex optimization, functional analysis, and statistical inference.