Proximal Subgradients, Marginal Values, and Augmented Lagrangians in Nonconvex Optimization

The Clarke subgradients of a nonconvex function p on Rⁿ are characterized in terms of limits of “proximal subgradients.” In the case where p is the optimal value function in a nonlinear programming problem depending on parameters, proximal subgradients correspond to saddlepoints of the augmented Lagrangian. When the constraint and objective functions are sufficiently smooth, this leads to a characterization of marginal values for a given problem in terms of limits of Lagrange multipliers in “neighboring” problems for which the standard second-order sufficient conditions for optimality are satisfied at a unique point.