<p>The paper is devoted to the study of some necessary and sufficient optimality conditions for a nonconvex extended-real-valued function having a global minimum/and a vector-valued mapping having a weakly efficient solution at a given point in terms of the augmented weak subdifferentials and the augmented normal cones in reflexive Banach spaces without any convexity assumption. By applying a special separation theorem for the nonconvex sets in reflexive Banach spaces, some necessary optimality conditions for a nonconvex (scalar/vector) function having a global minimum/and a (weakly) efficient solution concern the existence of a weakly subgradient pair are derived. Under some suitable assumptions, one of such conditions becomes the sufficient optimality condition respectively. An application of the obtained result for the nonsmooth nonconvex multiobjective mathematical programming problem having set, inequality and equality constraints is presented accordingly. </p>