The aim of the paper is to study the notions of normal cones and subdifferentials in the integral domain \({\mathbb{Z}}^{n}\) and derive optimality conditions for an abstract set constrained problem in terms of normal cones and subdifferentials. Various properties and calculus rules of normal cones, subdifferentials and singular subdifferentials are established and the differences with existing analogous rules in the real domain \({\mathbb{R}}^{n}\) are justified through counter examples.
