The purpose of this paper is to provide a survey of convergence and rate of convergence aspects of a cltass of recently proposed methods for constrained nfinimization--the, so-called, multiplier methods. The results discussed highlight the operational aspects of multiplier methods and demonstrate their significant advantages over ordinary penalty methods.

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.