On Weak and Strong Kuhn–Tucker Conditions for Smooth Multiobjective Optimization

We consider a smooth multiobjective optimization problem with inequality constraints. Weak Kuhn–Tucker (WKT) optimality conditions are said to hold for such problems when not all the multipliers of the objective functions are zero, while strong Kuhn–Tucker (SKT) conditions are said to hold when all the multipliers of the objective functions are positive. We introduce a new regularity condition under which (WKT) hold. Moreover, we prove that for another new regularity condition (SKT) hold at every Geoffrion‐properly efficient point. We show with an example that the assumption on proper efficiency cannot be relaxed. Finally, we prove that Geoffrion‐proper efficiency is not needed when the constraint set is polyhedral and the objective functions are linear.