Why does my solvers relax directive sometimes give a higher objective value than AMPLs relax_integrality option?
These two ways of relaxing integrality can give different results because they interact differently with AMPL’s presolve phase. Under option relax_integrality 1, all integer variables are changed to continuous before AMPL’s presolve. Thus presolve works on the “true” relaxation, and the reduced LP that comes out of presolve has the same objective value as the true relaxation. Under option relax_integrality 0, all integer variables remain integer through AMPL’s presolve phase. Presolve may take advantage of this integrality to further tighten bounds and reduce the problem size. If the solver’s relax directive is subsequently set, then it will solve the relaxation of the presolved integer program, which may not have the same objective value as the true relaxation. (Specifically, its value may be higher for a minimization, or lower for a maximization.) As a simple example, imagine a minimization model in which there are integer variables X[t] constrained by sum {t in 1..T} X[t] <> 7.5. If
