In this paper a notion called “directional Kuhn-Tucker condition” for quasidifferentiable programs with inequality constraints is introduced. This is a version of the Lagrange multiplier rule where the Lagrange multipliers depend on the directions. It is proved that this condition is a necessary condition for optimality. Under the assumption that the problem is directionally $\eta$-invex, it is also a sufficent condition for optimality. Some results on duality of the class of problems are obtained.