The Fritz-John (FJ) and KKT conditions are fundamental tools for characterizing minimizers and form the basis of almost all methods for constrained optimization. Since the seminal works of Fritz John, Karush, Kuhn and Tucker, FJ/KKT conditions have been enhanced by adding extra necessary conditions. Such an extension was initially proposed by Hestenes in the 1970s and later extensively studied by Bertsekas and collaborators. In this work, we revisit enhanced KKT stationarity for standard (smooth) nonlinear programming. We argue that every KKT point satisfies the usual enhanced versions found in the literature. Therefore, enhanced KKT stationarity only concerns the Lagrange multipliers. We then analyse some properties of the corresponding multipliers under the quasi-normality constraint qualification (QNCQ), showing in particular that the set of so-called quasinormal multipliers is compact if, and only if, QNCQ holds. In this sense, QNCQ plays the same role for enhanced KKT as the Mangasarian-Fromovitz constraint qualification plays for classical KKT. Also, we report some consequences of introducing an extra abstract constraint to the problem. Since enhanced FJ/KKT concepts are obtained by aggregating sequential conditions to FJ/KKT, we discuss the relevance of our findings to the so-called sequential optimality conditions, which have been crucial in generalizing the global convergence of a well-established safeguarded augmented Lagrangian method. Finally, we apply our theory to mathematical programs with complementarity constraints and multi-objective problems, improving and clarifying previous results from the literature.
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