Lagrange-type optimization with inequality constraints

In this paper, optimality conditions are derived for discrete and differential optimization of the Lagrange problem with inequality-type constraints. According to the proposed discretization method, we study problems with discrete-approximate inequalities. In this case, the transition from a discrete problem to a discrete-approximate problem is carried out by the equivalence theorem of subdifferential inclusions, which is the main tool used to establish optimality conditions for continuous problems. This approach plays a much more important role in deriving an adjoint discrete and differential inclusion generated by given inequality constraints. A numerical example demonstrates the effectiveness of the obtained theoretical results.