Extensions of the Kuhn-Tucker constraint qualification to generalized semi-infinite programming

This paper deals with the class of generalized semi- infinite programming problems ( GSIPs) in which the index set of the inequality constraints depends on the decision vector and all emerging functions are assumed to be continuously differentiable. We introduce two extensions of the Kuhn - Tucker constraint qualification ( which is based on the existence of a tangential continuously differentiable arc) to the class of GSIPs, prove a corresponding Karush - Kuhn - Tucker theorem, and discuss its assumptions. Finally, we present several examples which illustrate for the class of GSIPs some interrelations between the considered extensions of the Mangasarian - Fromovitz constraint qualification, the Abadie constraint qualification, and the Kuhn - Tucker constraint qualification.