Directional differentiability of the optimal value function in a nonlinear programming problem

A parameterized nonlinear programming problem is considered in which the objective and constraint functions are twice continuously differentiable. Under the assumption that certain multiplier vectors appearing in generalized second-order necessary conditions for local optimality actually satisfy the weak sufficient condition for local optimality based on the augmented Lagrangian, it is shown that the optimal value in the problem, as a function of the parameters, is directionally differentiable. The directional derivatives are expressed by a minimax formula which generalizes the one of Gol’shtein in convex programming.

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