Optimization of third-order discrete and differential inclusions described by polyhedral set-valued mappings

The present paper is concerned with the necessary and sufficient conditions of optimality for third-order polyhedral optimization described by polyhedral discrete and differential inclusions (PDIs). In the first part of the paper, the discrete polyhedral problem (PDIs) is reduced to convex minimization problem and the necessary and sufficient condition for optimality is derived. Then the necessary and sufficient conditions of optimality for discrete-approximation problem (P)_D are formulated using the transversality condition and approximation method for the continuous polyhedral problem (P)_C governed by PDI. On the basis on the obtained results in Section 3, we prove the sufficient conditions of optimality for the problem (P)_C. It turns out that the concerned method requires some special equivalence theorem, which allow us to make a bridge between (P)_D and (P)_C problems.