Generalized Bundle Methods

We study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties. This setting covers several algorithms from the literature that have been so far regarded as distinct. Under a different hypothesis on the stabilizing term and/or the function to be minimized, we prove finite termination, asymptotic convergence, and finite convergence to an optimal point, with or without limits on the number of serious steps and/or requiring the proximal parameter to go to infinity. The convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm, i.e., the management of the bundle of subgradients (-strategy) and of the proximal parameter (t-strategy). We extensively exploit a dual view of bundle methods, which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space, where nonlinear subproblems are approximately solved at each step with an inner li...

• Type: Journal
• Year: 2002
• Where: SIAMJO
• Authors: Antonio Frangioni
• Keywords: Bundle Methods, Closed Convex Function, Proximal Parameter, SIAMJO 2002