Sequential optimality conditions for optimization problems with additional abstract set constraints

Abstract

Recently, the so-called positive approximate Karush-Kuhn-Tucker sequential condition and the strict constraint qualification associated with thiscondition for general scalar problems with equality and inequality constraintswere introduced. In this work, we extend them to optimization problems with an 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 newconstraint qualifications and other constraint qualifications known in the literature as Abadie, quasi-normality and the approximate Karush-Kuhn-Tucker regularity constraint qualification. And finally, we introduce an AugmentedLagrangian Method for solving the optimization problem with abstract setconstraint and we show that it is possible to obtain global convergence underthe new condition.