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 (P-D) 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-DA) 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.