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The generalized Mangasarian-Fromowitz constraint qualification and optimality conditions for bilevel programs. (English) Zbl 1223.90061
The authors introduce a bilevel programming problem in terms of an upper level problem $$\min F(x, y) \quad\text{ s. t. } (x, y)\in X \times \Bbb R^m,\ y \in\Psi(x)\tag{ULP}$$ where $X$ is a closed subset of $\Bbb R^n$ 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 $F, f : \Bbb R^n \times \Bbb R^m \to\Bbb R$ 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 $G : \Bbb R^n \to \Bbb R^k$, $H : \Bbb R^n \to \Bbb R^l$, $g : \Bbb R^n\times \Bbb R^m \to \Bbb R^p$, $h : \Bbb R^n \times \Bbb R^m \to \Bbb R^q$ 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.

90C30Nonlinear programming
90C46Optimality conditions, duality
Full Text: DOI
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