I have taken some courses in Convex optimization. Now I would like to know a little bit more about the pure mathematical side. Is there any good books in convex analysis? I have read and worked with Boyds Convex Optimization book. Is there any video lectures anywhere? It would be nice as well. Thanks
I'm a big fan of the first 50 pages of Ekeland and Temam. It's a short, clear, beautiful explanation of the basics of convex analysis. I also like Rockafellar's books Convex Analysis, and also Conjugate Duality and Optimization.
Other books I recommend looking at: Introductory Lectures on Convex Optimization: A Basic Course by Nesterov, Convex Analysis and Nonlinear Optimization by Borwein and Lewis, Convex Analysis and Optimization by Bertsekas and Nedic, Convex Optimization Theory by Bertsekas, Nonlinear Programming by Bertsekas.
I've heard good things about the book Nonsmooth Analysis and Control Theory by Clarke.
There is a book by Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal called "Fundamentals of Convex Analysis" that I thought was great. It might not be exactly what you're looking for, as it goes through a lot of basics as well. But it's a very clear book, very easy to read despite being rigorous.