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**Optimality conditions for a class of mathematical programs with equilibrium constraints.**
*(English)*
Zbl 1039.90088

The paper deals with the following mathematical program: (MPEC)
\[
\min \{f(x,y) \mid y \in S(x), x \in W \}
\]
where \(S(x)= \{y \in \mathbb{R} ^n \mid F(x,y) \geq 0 (1)\), \(y \geq 0\), \(y^TF(x,y) =0\}. \) \(F(f)\) are assumed locally Lipschitz (continuously differentiable). The set of \((x,\;y)\) with \(y \in S(x)\) is denoted by \(\Omega\) hence feasible set of (MPEC) is described as \(\Omega \bigcap (W \times \mathbb{R}^n_+)\) and denoted by \(\Pi\):

\(y \in S(x)\) iff \(y \geq 0\) and \(y\) solves the generalized equation: \(0 \in F(x,y) + N_{ \mathbb{R}{^n}_+}(y)\), (where \(N_{ \mathbb{R}{^n}_+}(y)\), denotes the standard normal cone at \(y\) if \(y \geq 0\) and empty otherwise). In the paper \(K_{A}(z)\) denotes the generalized normal cone of the subset \(A\) at \(z\).

In this paper Mordukhovich’s generalized differential calculus is applied to derive necessary conditions in terms of upper approximations of the generalized normal cone of the feasible set of (MPEC), \(K_{\Pi}(y)\), and the subdifferential of \(f\), \( \partial^- f\). At first the author gives a constraint qualification condition (CQ), under which an upper approximation of \(K_{\Omega}(z)\) is obtained. It is proved that (CQ) is implied by linear independence constraint qualification condition (LICQ) and by the strong regularity of Robinson. Another constraint qualification (CQ*), which implies (CQ), is also proposed. It guarantees the required assumption [B. S. Mordukhovich, Approximation methods in problems of optimization and control. Nauka, Moscow (1988; Zbl 0643.49001)] to obtain \(K_{ A \cup B}(z) \subset K_A + K_B\), \(A,B \subset \mathbb{R}^n\). Applying that result and the above upper approximation of \(K_{\Omega}(z)\) , the elements of \(K_\Pi\) can be written in a more suitable way .

In the paper, different lemmas and propositions are also proven in order to express the constraints qualifications conditions and the results in a more workable form. Assuming that \(W\) is described by inequalities involving continuously differentiable functions and that Mangasarian -Fromowitz constraint qualification is fulfilled for \(W\), the author proves that the necessary and sufficient condition (CQ*) holds and that (CQ*) is implied by the so-called piecewise MFCQ. Besides if \(f\) is continuously differentiable it is remarked that the necessary optimality condition given in this paper is stronger that the one resulting of applying the generalized calculus of Clarke with the complementarity constraint replaced by the equality system: \[ \min \{F_i (x,y),x_i) \}, \quad i=1\dots n. \]

\(y \in S(x)\) iff \(y \geq 0\) and \(y\) solves the generalized equation: \(0 \in F(x,y) + N_{ \mathbb{R}{^n}_+}(y)\), (where \(N_{ \mathbb{R}{^n}_+}(y)\), denotes the standard normal cone at \(y\) if \(y \geq 0\) and empty otherwise). In the paper \(K_{A}(z)\) denotes the generalized normal cone of the subset \(A\) at \(z\).

In this paper Mordukhovich’s generalized differential calculus is applied to derive necessary conditions in terms of upper approximations of the generalized normal cone of the feasible set of (MPEC), \(K_{\Pi}(y)\), and the subdifferential of \(f\), \( \partial^- f\). At first the author gives a constraint qualification condition (CQ), under which an upper approximation of \(K_{\Omega}(z)\) is obtained. It is proved that (CQ) is implied by linear independence constraint qualification condition (LICQ) and by the strong regularity of Robinson. Another constraint qualification (CQ*), which implies (CQ), is also proposed. It guarantees the required assumption [B. S. Mordukhovich, Approximation methods in problems of optimization and control. Nauka, Moscow (1988; Zbl 0643.49001)] to obtain \(K_{ A \cup B}(z) \subset K_A + K_B\), \(A,B \subset \mathbb{R}^n\). Applying that result and the above upper approximation of \(K_{\Omega}(z)\) , the elements of \(K_\Pi\) can be written in a more suitable way .

In the paper, different lemmas and propositions are also proven in order to express the constraints qualifications conditions and the results in a more workable form. Assuming that \(W\) is described by inequalities involving continuously differentiable functions and that Mangasarian -Fromowitz constraint qualification is fulfilled for \(W\), the author proves that the necessary and sufficient condition (CQ*) holds and that (CQ*) is implied by the so-called piecewise MFCQ. Besides if \(f\) is continuously differentiable it is remarked that the necessary optimality condition given in this paper is stronger that the one resulting of applying the generalized calculus of Clarke with the complementarity constraint replaced by the equality system: \[ \min \{F_i (x,y),x_i) \}, \quad i=1\dots n. \]

Reviewer: S. M. Allende-Alonso (Ciudad Habana)