A result in surrogate duality for certain integer programming problems

The authors consider linear programming problems with some equality constraints. For such problems, surrogate relaxation formulations relaxing equality constraints exist with zero primal-dual gap both when all variables are restricted to be integers and when no variable is required to be integer. However, for such surrogate formulations, when the variables are mixed-integer, the primal-dual gap may not be zero. The authors establish this latter result by a counterexample.