Information theory and convex optimization 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.
The books below may be more to your liking, but in general, the texts/lecture notes are written for the use of (mainly) postgraduate students in engineering and cannot presume deep knowledge of convex analysis. 


Csizsar, I., and Korner, J., Information Theory: Coding Theorems for Discrete Memoryless Systems, 2nd Ed, Cambridge.
Berger, T., Rate Distortion Theory, quite old probably late 70's or early 80s, can't remember the publisher maybe Wiley.
Yeung, R.W., A First Course in Information Theory, Springer.


The research articles on Shannon theory and related fields in, say, IEEE Transactions on Information Theory, may fit the bill better, though not always. 

An older text which may also be of interest is 

Wolfowitz, J., Coding Theorems of Information Theory, Springer, 1960's.