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The positive approximate Karush-Kuhn-Tucker sequential condition and the strict constraint qualification associated with this condition for general scalar problems with equality and inequality constraints have recently been introduced.In this paper, we extend them to optimization problems with additional abstract set constraints.We also present an extension of the approximate Karush-Kuhn-Tucker sequential condition and its related strict constraint qualification.Furthermore, we explore the relations between the new constraint qualification and other constraint qualifications known in the literature as Abadie, quasi-normality and the approximate Karush-Kuhn-Tucker regularity constraint qualification.Finally, we introduce an augmented Lagrangian method for solving the optimization problem with abstract set constraints and we show that it is possible to obtain global convergence under the new condition.
Fazzio et al. (Tue,) studied this question.