Generalized Bundle Methods

Generalized Bundle Methods

We study a class of generalized bundle methods where 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 different hypothesis on the stabilizing term and/or the function to be minimized, we prove finite termination, asymptotic convergence and finite conv...

Dynamic bundle methods

Lagrangian relaxation is a popular technique to solvedifficult optimization problems. However, the applicability of this technique depends on having a relatively low number of hard constraints to dualize. When there are many hard constraints, it may be preferable to relax them dynamically, according to some rule depending on which multipliers are active. From the dual point of view, this approa...

Functional Bundle Methods

Recently, gradient descent based optimization procedures and their functional gradient based boosting generalizations have shown strong performance across a number of convex machine learning formulations. They are particularly alluring for structured prediction problems due to their low memory requirements [5], and recent theoretical work has show that they converge fast across a wide range of ...

Inexact Dynamic Bundle Methods

We give a proximal bundle method for minimizing a convex function f over R+. It requires evaluating f and its subgradients with a possibly unknown accuracy ε ≥ 0, and maintains a set of free variables I to simplify its prox subproblems. The method asymptotically finds points that are ε-optimal. In Lagrangian relaxation of convex programs, it allows for ε-accurate solutions of Lagrangian subprob...

2 BUNDLE METHODS FOR SDPproblems