On optimality conditions in nonsmooth inequality constrained minimization

First order necessary optimality conditions for a minimum of an inequality constrained minimization problem are given in terms of approximate quasidifferentials, without the usual differentiability, convexity or locally Lipschitz assumptions. The main result is obtained with the help of a semi-infinite Gordan type alternative theorem. Sufficient conditions for a minimum are also given with the usual convexity assumption replaced by an invex condition.