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.

We use a primal-dual scheme to devise a new update rule for a penalty function method applicable to general optimization problems, including nonsmooth and nonconvex ones. The update rule we introduce uses dual information in a simple way. Numerical test problems show that our update rule has certain advantages over the classical one. We study the relationship between exact penalty parameters and dual solutions. Under the differentiability of the dual function at the least exact penalty parameter, we establish convergence of the minimizers of the sequential penalty functions to a solution of the original problem. Numerical experiments are then used to illustrate some of the theoretical results. Key words: Penalty function method, penalty parameter update, least exact penalty parameter, duality, nonsmooth optimization, nonconvex optimization. Mathematical Subject Classification: 49M30; 49M29; 49M37; 90C26; 90C30. 1