Once the main ideas on cutting-plane methods have been presented in the previous chapter, here we go further to present state-of-the-art algorithms for minimizing a non-smooth convex function f over a “simple” feasible set \(X \subset \mathbb {R}^n\) . This chapter goes deep into the theory of bundle methods, shedding light on the intuition behind each variant, their specificities, strengths, and weaknesses. For didactic purposes, we provide abridged versions of the considered bundle methods before presenting their more efficient forms capable of exploiting the possible additive structure of the objective function and even inexact oracles. As in the previous chapter, f may only be assessed by a first-order (inexact) oracle.