The paper is mainly devoted to the theory of duality of boundary value problems (BVPs) for differential inclusions of higher orders. For this, on the basis of the apparatus of locally conjugate mappings in the form of Euler-Lagrange-type inclusions and transversality conditions, sufficient optimality conditions are obtained. Wherein remarkable is the fact that inclusions of Euler-Lagrange type for prime and dual problems are "duality relations". To demonstrate this approach, the optimization of some third-order semilinear BVPs and polyhedral fourth-order BVPs is considered. These problems show that sufficient conditions and dual problems can be easily established for problems of any order.