Optimality conditions for vector mathematical programming via a theorem of the alternative

This paper is concerned with the optimality conditions for nonsmooth and nonconvex vector mathematical programming but with the existence of the Hadamard directional derivatives of objective and constraint functions. By using a generalization of the Gordan alternative theorem proved by Jeyakumar, we establish the optimality conditions in terms of Lagrange–Kuhn–Tucker multipliers. Also the sufficient optimality conditions are given whenever the data are convex. In the last, we give an application to vector fractional mathematical programming.