In Optimization Theory, necessary and sufficient optimality conditions play an essential role. They allow, first of all, checking whether a point under study satisfies the conditions, and, secondly, if it does not, finding a "better" point. For the class of directionally differentiable functions, a necessary condition for an unconstrained minimum requires 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 of inclusion. The problem of verifying this condition is reduced to that of finding the point of some convex and compact set C which is 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 method. In the classical Chebyshev polynomial approximation problem, necessary optimality conditions are expresse