The present paper deals with the theory of duality for the Mayer problem given by second-order polyhedral discrete and differential inclusions. First, we formulate the conditions of optimality in the form of Euler-Lagrange type inclusions and transversality conditions for the polyhedral problems with second-order discrete and differential inclusions. Second, we establish dual problems for discrete and differential inclusions based on the infimal convolution concept of convex functions and prove the results of duality. For both primary and dual problems, the Euler-Lagrange type inclusions are "duality relations" and that the dual problem for discrete-approximate problems bridges the gap between the dual problems of discrete and continuous problems. As a result, the passage to the limit in the dual problem with discrete approximations plays a substantial role in the following investigations, without any which can hardly ever be established by any duality to the continuous problem. Furthermore, the numerical results also are provided.