In solving optimization problems, necessary and sufficient optimality conditions play an outstanding role. They allow, first of all, to check whether a point under study satisfies the conditions, and, secondly, if it does not, to find a “better” point. This is why such conditions should be “constructive” letting to solve the above-mentioned problems. For the class of directionally differentiable functions in, a necessary condition for an unconstrained minimum requires for the directional derivative to be non-negative in all directions. This condition becomes efficient for special classes of directionally differentiable functions. For example, in the case of convex and max-type functions, the necessary condition for a minimum takes the form where is a convex compact set. The problem of verifying this condition is reduced to that of finding the point of C which is the nearest to the origin. If the origin does not belong to C, we easily find the steepest descent direction and are able to construct a numerical me