TITLE: Learning about convex optimisation
QUESTION [7 upvotes]: I'm interested in learning a bit about convex optimisation. The wikipedia article contains the following paragraph:

The convexity of $f$ makes the powerful tools of convex analysis
  applicable. In finite-dimensional normed spaces, the Hahn–Banach
  theorem and the existence of subgradients lead to a particularly
  satisfying theory of necessary and sufficient conditions for
  optimality, a duality theory generalizing that for linear programming,
  and effective computational methods.

which I find fascinating. It's telling me that I can use advanced mathematical tools to study optimisation, much like the article explaining how Lagrange multipliers come from manifolds.
I'm looking for a book gives rigorous proofs of the mathematical statements in the quote.
People recommended Boyd and Vandenberghe, but it's not the hi-tech exposition I'm looking for.

REPLY [2 votes]: I highly recommend the following book:
Hiriart-Urruty, Jean-Baptiste, and Claude Lemaréchal. Fundamentals of convex analysis.
It is a rigorous book on the mathematical foundations on convex analysis, and is designed for learning the topic from the ground up. I can't comment on how the book compares to Bersekas' Convex Optimization Theory, since I haven't read his book. Probably both  would be good choices, Bertsekas is a great author.
For algorithms, a (very) new option is "Convex Optimization Algorithms," also by Bertsekas. I've only read part of it so far, but I've enjoyed it a lot.