Optimality Conditions For Higher Order Polyhedral Discrete And Differentıal Inclusions

Abstract

The problems considered in this paper are described in polyhedral multi-valued mappings for higher-order(s-th) discrete $(PDSIs)$ and differential inclusions $(PDFIs)$. The present paper focuses on the necessary and sufficient conditions of optimality for the optimization of these problems. By converting the $PDSIs$ problem into a geometric constraint problem, we formulate the necessary and sufficient conditions of optimality for a convex minimization problem with linear inequality constraints. Then, in terms of the Euler-Lagrange type $PDSIs$ and the specially formulated transversality conditions, we are able to obtain conditions of optimality for the $PDSIs$. In order to obtain the necessary and sufficient conditions of optimality for the discrete-approximation problem $PDSIs$, we reduce this problem to the form of a problem with higher-order discrete inclusions. Finally, by formally passing to the limit, we establish the sufficient conditions of optimality for the problem with higher-order $PDFIs$. A numerical approach is developed to solve a polyhedral problem with second-order polyhedral discrete inclusions.