Saddlepoints in group and semigroup minimization

The group or semigroup minimization problem, derived from integer programming, is discussed. A dual form of this problem is stated and weak and strong duality theorems, together with complementarity conditions, are shown. Moreover, a Lagrangean function is introduced and it is shown that the classical saddlepoint theorems still hold good. The objective function of the minimization problem is formed using elements drawn from an ordered d-monoid, thereby treating sum, bottleneck and lexicographic objectives from a unified point of view.