Mathematical Programming solvers typically work with finite-precision numbers, which
leads to concerns on numerical stability.
While general numerical stability is a well-established topic, below
we highlight some common points for non-smooth models.
Importance of tight bounds: “big-M” constraints
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, for example,
an implication
may be linearized as
with the big-M constant taken as the upper bound on \(x\).
Thus, big-M constraints require finite bounds on participating variables.
They should be as tight as possible, ideally between \(\pm10^4\).
In any case, 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 (use with caution).
In some cases, a different formulation can help automatic detection
of bounds. In the below example, the ==> / else operator
cannot do this for variables DL:
var DL {l in Lset, s in Sset};
subject to setDL {l in Lset, s in Sset}:
L[l,s] > K[l,s] ==> DL[l,s] = L[l,s] else DL[l,s] = 0;
Reformulating this constraint using if-then enables automatic
bound deduction:
subject to setDL {l in Lset, s in Sset}:
DL[l,s] = if L[l,s] > K[l,s] then L[l,s];
Piecewise-linear functions
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 \(\pm10^4\) (for approximated nonlinear functions,
hard bounds of up to \(\pm10^6\) 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.