The MOSEK Python optimizer API manual Version 7.0 (Revision 141)

178 CHAPTER 14. PRIMAL FEASIBILITY REPAIR
which is infeasible for any ɛ ≠ 0. Here the infeasibility is caused by a linear dependency in the constraint
matrix and that the right-hand side does not match if ɛ ≠ 0. Observe even if the problem is feasible
then just a tiny perturbation to the right-hand side will make the problem infeasible. Therefore, even
though the problem can be repaired then a much more robust solution is to avoid problems with linear
dependent constraints. Indeed if a problem contains linear dependencies then the problem is either
infeasible or contains redundant constraints. In the above case any of the equality constraints can be
removed while not changing the set of feasible solutions.
To summarize linear dependencies in the constraints can give rise to infeasible problems and therefore
it is better to avoid them. Note that most network flow models usually is formulated with one linear
dependent constraint.
Next consider the problem
minimize
subject to x 1 − 0.01x 2 = 0
x 2 − 0.01x 3 = 0
x 3 − 0.01x 4 = 0
x 1 ≥ − 1.0e − 9
x 1 ≤ 1.0e − 9
x 4 ≤ − 1.0e − 4
(14.2)
Now the MOSEK presolve for the sake of efficiency fix variables (and constraints) that has tight bounds
where tightness is controlled by the parameter dparam.presolve tol x. Since, the bounds
−1.0e − 9 ≤ x 1 ≤ 1.0e − 9
are tight then the MOSEK presolve will fix variable x 1 at the mid point between the bounds i.e. at 0.
It easy to see that this implies x 4 = 0 too which leads to the incorrect conclusion that the problem is
infeasible. Observe tiny change of the size 1.0e-9 make the problem switch from feasible to infeasible.
Such a problem is inherently unstable and is hard to solve. We normally call such a problem ill-posed.
In general it is recommended to avoid ill-posed problems, but if that is not possible then one solution
to this issue is is to reduce the parameter to say dparam.presolve tol x to say 1.0e-10. This will at
least make sure that the presolve does not make the wrong conclusion.
14.2 Automatic repair
In this section we will describe the idea behind a method that automatically can repair an infeasible
probem. The main idea can be described as follows.
Consider the linear optimization problem with m constraints and n variables
which is assumed to be infeasible.
minimize
c T x + c f
subject to l c ≤ Ax ≤ u c ,
l x ≤ x ≤ u x ,
(14.3)