This paper focuses on information which is lacking in the straightforward application of subgradient optimization methods and on how to access it through suitable modifications or extensions of the scheme. A technique which is both classical and modern for inducing properties which an original sequence lacks is the construction of an ergodic (averaged) sequence which smoothes out oscillations in the original sequence. We introduce the computation of ergodic sequences from sequences of subgradients, normal elements, and Lagrange

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. Key words: Nonconvex programming; nonsmooth optimization; augmented Lagrangian; sharp Lagrangian; subgradient optimization.

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