Mathematical Programming solvers typically work with finite-precision numbers, which leads to concerns on numerical stability.
The MP library simplifies relational operators into “indicator” constraints. Solvers natively supporting indicators, usually handle them in a numerically stable way. Otherwise, they have to be linearized by the so-called “big-M” constraints. The big-M constants require finite bounds on expressions. For numerical stability these bounds should not exceed the reciprocal of the integrality tolerance (option inttol). A default big-M value can be set with the option cvt:bigM.
Piecewise-linear expressions can be modeled in AMPL directly, or arise from approximations of other functions. Solvers which support PL expressions, usually handle them algorithmically in a numerically stable way. Otherwise, if PL expressions are linearized, it is recommended to have the argument and result variables bounded in [-1e+4, 1e+4] (for approximated nonlinear functions, hard bounds of up to [-1e+6, 1e+6] are imposed). The stability can be improved in some cases by decreasing integer tolerance, Gurobi’s intfocus and numfocus options, switching off presolve in the solver, and other tuning measures.