In this paper we study the existence of solutions for a type of nonlocal minimization problem. The minimization principle is relaxed by means of Young measures, and different generalized necessary conditions of optimality are used: the Euler-Lagrange equation and the extended version of the Weierstrass minimum principle. These conditions are applied to establish some results concerning the existence and the description of the simplest relaxation of the problem in terms of Young measures. The main result of the paper is the description of the simplest relaxation by considering only a combination of two Dirac measures. The research is focused on the homogeneous and nonhomogeneous 1-dimensional scalar cases. The results obtained are numerically exploited to compute the lower semicontinuous envelope for some academic examples. A detailed numerical analysis of the nonlocal regularization for Young's tacking problem formulated in [D. Brandon and R. Rogers, Appl. Math. Optim., 25 (1992), pp. 287-301] is given.