We give a proximal bundle method for minimizing a convex function $f$ over $\mathbb{R}_+^n$. It requires evaluating $f$ and its subgradients with a possibly unknown accuracy $\epsilon\ge0$, and maintains a set of free variables $I$ to simplify its prox subproblems. The method asymptotically finds points that are $\epsilon$-optimal. In Lagrangian relaxation of convex programs, it allows for $\epsilon$-accurate solutions of Lagrangian subproblems, and finds $\epsilon$-optimal primal solutions. For programs with exponentially many constraints, it adopts a relax-and-cut approach where the set $I$ is extended only if a separation oracle finds a sufficiently violated constraint. In a simplified version, each iteration involves solving an unconstrained prox subproblem. For semidefinite programming problems, we extend the spectral bundle method to the case of weaker conditions on its optimization and separation oracles, and on the original primal problem.

## Citation

Krzysztof C. Kiwiel, "Inexact Dynamic Bundle Methods, Tech. report, Systems Research Institute, Warsaw, December 2010