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.
Importance of tight bounds: “big-M” constraints
Try & keep your variable and constraint bounds tight.
For example, 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 easily 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 \(\pm10^4\) (the tighter the better;
for approximated nonlinear functions,
hard bounds of up to \(\pm10^6\) are imposed by default). 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.