A general discrete optimization problem is investigated that includes integer polynomial programs as special cases. To exploit the discrete monotonic structure of these problems, a special class of cuts called monotonicity cuts are developed and then adjusted according to a suitable procedure to accommodate discrete requirements. As illustration, the method is applied to solve a discrete location problem which is also a variant of the well known engineering problem of design centering. Computational results are reported for instances of the latter problem with up to 100 variables and 500 constraints.