Mingling: Mixed-Integer Rounding with Bounds

Mixed-integer rounding (MIR) is a simple, yet powerful procedure for generating valid inequalities for mixed-integer programs. When used as cutting planes, MIR inequalities are very effective for problems with unbounded integer variables. For problems with bounded integer variables, however, cutting planes based on lifting techniques appear to be more effective. This is not surprising as lifting techniques make explicit use of the bounds on variables, whereas the MIR procedure does not.

In this paper we describe a simple procedure, which we call mingling, for incorporating variable bound information into mixed-integer rounding. By explicitly using the variable bounds, the mingling procedure leads to strong inequalities for mixed-integer sets with bounded variables. We show that facets of the mixed-integer knapsack sets derived earlier by superadditive lifting techniques are mingling inequalities. In particular, the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of Marchand and Wolsey [9] as well as the continuous integer knapsack cover and pack inequalities of Atamtürk [1, 3]. In addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Günlük [7] under some conditions.