The Value function of a Mixed-Integer Linear Program with a Single Constraint

The value function of a mixed-integer linear program (MILP) is a function that returns the optimal solution value as a function of the right-hand side. In this paper, we analyze the structure of the value function of a MILP with a single constraint. We show that the value function is uniquely determined by a finite number of break points and at most two slopes. We derive conditions for the value function to be continuous and analyze its behavior where it is discontinuous. We also propose a method for systematically extending the value function from a neighborhood of the origin to the entire real line using the technique of maximal subadditive extension. ∗Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015, [email protected], http://coral.ie.lehigh.edu/ ̃menal †Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015, [email protected], http://www.lehigh.edu/ ̃tkr2