Generalized Bundle Methods

We propose a class of generalized Bundle methods where the stabilizi= term can be any closed convex function satisfying certain properties. = We prove finite termination, asymptotic convergence and finite convergenc= e to an optimal point of different variants of the algorithm: for some of= these proofs, f-regularity of the function (a new generalization of inf-= compactness) is required. Our analysis gives a unified convergence proof = for several classes of Bundle methods that have been so far regarded as d= istinct, enhancing on the results known for some of them: furthermore, it= covers methods proposed in the literature that had not previously been r= ecognised as Bundle methods. A novelty in our approach is the proposal of= a dual for the minimization problem: we show that Bundle methods can be = seen as a dual ascent approach to one single nonlinear problem in the dua= l space, where nonlinear subproblems are approximately solved at each ste= p with an inner linearization approach. This interpretation is particular= ly