For combinatorial or discrete optimization, AMPL has always provided the option of including integer-valued variables in algebraic objectives and constraints. This is sufficient to make good use of mixed-integer programming solvers that use a classical branch-and-bound procedure.

Optimization systems for constraint programming (CP) also use a branching or tree search, but are able to work with a much broader variety of combinatorial expressions. In exchange for this advantage, CP solvers typically forego the ability to generate the bounds and cuts that are used to great advantage to limit the search in integer programming. The CP approach thus tends to be advantageous for problems that are “highly” combinatorial, in the sense that their formulation in terms of integer variables is artificial and hard to work with, or requires a very large number of variables or constraints, or fails to yield strong bounds.