The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs
The authors introduce a bilevel programming problem in terms of an upper level problem \[ \min F(x, y) \quad\text{ s. t. } (x, y)\in X \times \mathbb R^m,\;y \in\Psi(x)\tag{ULP} \] where \(X\) is a closed subset of \(\mathbb R^n\) and \(\Psi(x)\) is the solution set of the lower level problem defined as: \[ \min f(x, y)\quad\text{ s. t. }y \in K(x). \] The functions \(F, f : \mathbb R^n \times \mathbb R^m \to\mathbb R\) are continuous, \(X = \{ x \mid G(x) \leq 0, H(x) = 0\}\), \(K(x) = \{ y\mid g(x, y) \leq 0, h(x, y) = 0\}\), where the functions \(G : \mathbb R^n \to \mathbb R^k\), \(H : \mathbb R^n \to \mathbb R^l\), \(g : \mathbb R^n\times \mathbb R^m \to \mathbb R^p\), \(h : \mathbb R^n \times \mathbb R^m \to \mathbb R^q\) are all continuous. Assuming feasibility of (ULP) the optimal value reformulation is \[ \min F(x, y)\quad \text{ s. t. } x \in X, \;y\in K(x),\;f(x, y) \leq \phi(x)\tag{OV} \] where \(\phi(x)\) is defined as: \(\phi(x) = \min \{ f(x, y) \mid y \in K(x)\}\). The authors contend that the problems (ULP) and (OV) are locally and globally equivalent. It is shown that the Mangasarian-Fromowitz constraint qualification in terms of basic generalized differentiation constructions of Mordukhovich fails and a weakened form of the same is used to derive Karush-Kuhn-Tucker optimality conditions for (ULP). They also suggest new sufficient conditions for the partial calmness based on a more general notion of the weak sharp minimum.
scientific article; zbMATH DE number 5864225
scientific article