## Directional differentiability of the optimal value function in a nonlinear programming problem.(English)Zbl 0546.90088

A parameterized minimization problem is considered in which all the functions involved (the objective function and the constraint ones) are of class $$C^ 2$$. Under the assumption that certain multiplier vectors arising in the derivation of second-order necessary conditions for optimality satisfy the ad hoc condition for local optimality based on the augmented Lagrangian, it is shown that the marginal function in the problem, considered as a function of the parameters, has a usual directional derivative. The directional derivatives are expressed by a min-max formula.

### MSC:

 90C31 Sensitivity, stability, parametric optimization 49K10 Optimality conditions for free problems in two or more independent variables 49M37 Numerical methods based on nonlinear programming 90C30 Nonlinear programming
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