2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229977We consider integer programming and the semi-group membership problem. We provide the following theorem of the alternative: the system Ax=b has no nonnegative integral solution x if and only if p(b) <0 for some given polynomial p whose vector of coefficients lies in a convex cone that we characterize. We also provide a hierarchy of linear programming relaxations, where the continuous case Ax=b with x real and nonnegative, describes the first relaxation in the hierarchy.21 pagesOptimization and ControlDiscrete Mathematics90, C10text