Preprint A652/2010
An inexact modified subgradient algorithm for primal-dual problems via augmented Lagrangians

Jefferson Melo | Burachik, Regina | Iusem, Alfredo

**Keywords: **
Augmented Lagrangian | inexact subgradient algorithm

We consider a primal optimization problem in a reflexive Banach space, and a duality scheme via general augmented Lagrangians. For solving the dual problem, we introduce and analyze a new parametrized inexact modified subgradient algorithm, which generates a primal-dual sequence, and we focus on two simple new choices for the stepsize. We prove that any weak accumulation point of the primal sequence is a primal solution, and that the dual sequence converges weakly to a dual solution, as long as the dual optimal set is nonempty. Moreover, we establish primal convergence even when the dual optimal set is empty. Our second choice of the stepsize gives rise to a variant of the method which has finite termination.