Let \(\mathbb{X}\) be a Banach space with the topological dual \(\mathbb{X}^*\) equipped with the weak \(*\) topology and \(\mathbb{Y}\) be an \(m\)-dimensional Hilbert space over \(\mathbb{R}\). This paper analyzes the following mathematical program \[ \begin{aligned} (\text{MPGC}) \hskip1truecm &\min\limits_{x\in\Omega} f(x) \\ &\text{s. t.} F(x) \in \Lambda,\end{aligned} \]
where \( f : \mathbb{X} \to \mathbb{R}\) and \(F: \mathbb{X} \to \mathbb{Y} \) are Lipschitzian near the point of interest and \(\Omega\) and \(\Lambda\) are nonempty and closed subsets of \(\mathbb{X}\) and \(\mathbb{Y}\), respectively. The problem (MPGC) includes as special cases the classical nonlinear program, the cone constrained program, the mathematical program with equilibrium constraint, the semidefinite program, the mathematical program with semidefinite cone complementarity constraints, among others.
The authors obtain the nonsmooth enhanced Fritz John necessary optimality conditions in terms of the approximate subdifferential. In the case where the Banach space is a weakly compactly generated Asplund space, the optimality condition obtained are expressed in terms of the limiting subdifferential, while in the general case they are expressed in terms of the Clarke subdifferential. Afterwards, the enhanced Karush-Kuhn-Tucker condition under the pseudo-normality and the quasi-normality conditions which are weaker than the classical normality conditions, are also obtained. It is proven that the quasi-normality is a sufficient condition for the existence of local error bounds of the constraint system. Finally, a tighter upper estimate for the subdifferentials of the value function of the perturbed problem in terms of the enhanced multipliers, is established.