Augmented Lagrangians were first employed in an algorithm for solving nonlinear programming problems with equality constraints. However, the approach was soon extended to inequality constraints and shown in the case of convex programming to correspond to applying the proximal point algorithm to solve a dual problem. Recent developments make it possible now to articulate that ALM approach in extensions far beyond classical nonlinear programming. This is tied to revelations of a kind local dual problem, based on advances in understanding second-order sufficient conditions for local optimality. Surprising insights about stepsizes are obtained even for the classical NLP implementation.

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