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 mixed-integer programming 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 MIR. 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 mixed-integer knapsack sets derived earlier by superadditive lifting techniques can be obtained by the mingling procedure. In particular, the mingling inequalities developed in this paper subsume the continuous cover and reverse continuous cover inequalities of Marchand and Wolsey (Math Program 85:15-33, 1999) as well as the continuous integer knapsack cover and pack inequalities of Atamturk (Math Program 98:145-175, 2003; Ann Oper Res 139:21-38, 2005). In addition, mingling inequalities give a generalization of the two-step MIR inequalities of Dash and Gunluk (Math Program 105:29-53, 2006) under some conditions.