Augmented Lagrangians and Marginal Values in Parameterized Optimization Problems

Rockafellar, R.T. (1981). Augmented Lagrangians and Marginal Values in Parameterized Optimization Problems. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-81-019

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Abstract

When an optimization problem depends on parameters, the minimum value in the problem as a function of the parameters is typically far from being differentiable. Certain subderivatives nevertheless exist and can be intepreted as generalized marginal values. In this paper such subderivatives are studied in an abstract setting that allows for infinite dimensionality of the decision space. By means of the notion of proximal subgradients, a new general formula of subdifferentiation is established which provides an upper bound for the marginal values in question and a very broad criterion for local Lipschitz continuity of the optimal value function. Augmented Lagrangians are introduced and shown to lead to still sharper estimates in terms of special multiplier vectors. This approach opens a way to taking higher-order optimality conditions into account in such estimates.

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