A Unified Augmented Lagrangian Approach to Duality and Exact Penalization

In this paper, the existence of an optimal path and its convergence to the optimal set of a primal problem of minimizing an extended real-valued function are established via a generalized augmented Lagrangian and corresponding generalized augmented Lagrangian problems, in which no convexity is imposed on the augmenting function. These results further imply a zero duality gap property between the primal problem and the generalized augmented Lagrangian dual problem. A necessary and sufficient condition for the exact penalty representation in the framework of a generalized augmented Lagrangian is obtained. In the context of constrained programs, we show that generalized augmented Lagrangians present a unified approach to several classes of exact penalization results. Some equivalences among exact penalization results are obtained.

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