Preprint A728/2013
The exact penalty map for nonsmooth and nonconvex optimization
Jefferson Melo | Burachik, Regina | Iusem, Alfredo
Keywords: Duality scheme | exact penalty | augmented Lagrangian | nonconvex optimization | Nonsmooth optimization | Banach spaces
Augmented Lagrangian duality provides zero duality gaps and saddle point properties for nonconvex optimization. On the basis of this duality, subgradient-like methods can be applied to the (convex) dual of the original problem. These methods usually recover the optimal value of the problem, but may fail to provide a primal solution. We prove that the recovery of the primal solution by such methods can be characterized in terms of (i) the differentiability properties of the dual function and (ii) the exact penalty properties of the primal-dual pair.We also connect the property of finite termination with exact penalty properties of the primal-dual pair. In order to establish these facts, we associate the primal dual pair to a penalty map. This map, which we introduce here, is a convex and globally Lipschitz function, and its epigraph encapsulates information on the primal and dual solution sets.