Sequence Independent Lifting for Mixed-Integer Programming

Lifting is a procedure for deriving strong valid inequalities for a closed set from inequalities that are valid for its lower dimensional restrictions. It is arguably one of the most effective ways of strengthening linear programming relaxations of 0–1 programming problems. Wolsey (1977) and Gu et al. (2000) show that superadditive lifting functions lead to sequence independent lifting of valid inequalities for monotone 0–1 programming and for monotone mixed 0–1 programming, respectively. We show that this property holds for general mixed-integer programming (MIP) as well if lower dimensional restrictions are obtained by setting integer variables to a bound. Lifting with general integer variables is computationally harder than lifting with 0–1 variables, because the former requires the solution of nonlinear integer problems rather than linear integer problems. Here we see that nonlinearity in lifting problems is resolved easily with superadditive lifting functions. The results presented here may pave the way for efficient applications of lifting with general integer variables.