On enhanced KKT optimality conditions for smooth nonlinear optimization

The Fritz-John (FJ) and KKT conditions are a fundamental tool to characterize minimizers and lie in the root of almost any method 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 first proposed by Hestenes in the 1970s and later extensively studied by Bertsekas and collaborators. In this work we revisited enhanced KKT stationarity for standard (smooth) nonlinear programming. We prove that every KKT point satisfies usual enhanced versions from the literature. Therefore, enhanced KKT stationarity only concerns the Lagrange multipliers. We then analyse some properties of such improved multipliers with quasi-normality (QN), showing in particular that the set of them is compact if, and only if, QN holds. In this sense, QN has the same status for enhanced KKT that Mangasarian-Fromovitz constraint qualification has for classical KKT. Also, we report some consequences of adding an extra abstract constraint to the problem. As enhanced FJ/KKT concepts are obtained by aggregating sequential conditions to FJ/KKT, we report the relevance of our results for the so-called sequential optimality conditions, that have been crucial to generalize global convergence of a consolidate 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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