I'm taking a graduate level course in information theory and I'm constantly struck by how much convex optimization there is in this subject. However, the proofs seem to shy away from using the full machinery of relaxation theory, duality, etc. This is understandable since you don't want to require a full semester of convex optimization in order to teach this stuff. But as someone fairly well-versed in optimization, I feel like I'm missing out on a lot of elegance and intuition when these links aren't explored more. I often notice proofs that would be way shorter if you had utilized convex analysis as well.
Are there books that cover information theory more from this perspective? We're mostly using lecture notes from Stefan Moser, Y. Polyanskiy and Y. Wu, as well as Network Information Theory by El Gamal.