Parametrically Robust Optimality in Nonlinear Programming*

In nonlinear programming, the strong second-order optimality condition and the linearly independent gradient condition have many uses. In particular, the first guarantees that a point is an isolated locally optimal solution, while the second insures the uniqueness of the associated multiplier vector, but other, less stringent assumptions would already be enough for that. In fact, the combination of these two conditions is equivalent to having a powerful stability property beyond just local primal-dual uniqueness. That property is parametric robustness: almost no matter how parameters are introduced into the objective and constraint functions, the dependence of the locally optimal solution and its multiplier vector on the parameters will be single-valued and Lipschitz continuous. Then, moreover, the mapping from parameter vector to solution-multiplier pair will have directional derivatives which meet the high standards of semidifferentiability. At any point and in any direction, the derivative can be calculated by solving an auxiliary problem of quadratic programming.