Dual applications of proximal bundle methods, including Lagrangian relaxation of nonconvex problems

We exhibit useful properties of proximal bundle methods for finding min(S) f, where f and S are convex. We show that they asymptotically find objective subgradients and constraint multipliers involved in optimality conditions, multipliers of objective pieces for max-type functions, and primal and dual solutions in Lagrangian decomposition of convex programs. When applied to Lagrangian relaxation of nonconvex programs, they find solutions to relaxed convexified versions of such programs. Numerical results are presented for unit commitment in power production scheduling.