Approximate Karush–Kuhn–Tucker Condition in Multiobjective Optimization

We extend the so-called approximate Karush-Kuhn-Tucker condition from a scalar optimization problem with equality and inequality constraints to a multiobjective optimization problem. We prove that this condition is necessary for a point to be a local weak efficient solution without any constraint qualification, and is also sufficient under convexity assumptions. We also state that an enhanced Fritz John-type condition is also necessary for local weak efficiency, and under the additional quasi-normality constraint qualification becomes an enhanced Karush-Kuhn-Tucker condition. Finally, we study some relations between these concepts and the notion of bounded approximate Karush-Kuhn-Tucker condition, which is introduced in this paper.