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

Chapter 14
Primal feasibility repair
Section 13.2.2 discusses how MOSEK treats infeasible problems. In particular, it is discussed which
information MOSEK returns when a problem is infeasible and how this information can be used to
pinpoint the cause of the infeasibility.
In this section we discuss how to repair a primal infeasible problem by relaxing the constraints in a
controlled way. For the sake of simplicity we discuss the method in the context of linear optimization.
14.1 Manual repair
Subsequently we discuss an automatic method for repairing an infeasible optimization problem. However,
it should be observed that the best way to repair an infeasible problem usually depends on what
the optimization problem models. For instance in many optimization problem it does not make sense
to relax the constraints x ≥ 0 e.g. it is not possible to produce a negative quantity. Hence, whatever
automatic method MOSEK provides it will never be as good as a method that exploits knowledge
about what is being modelled. This implies that it is usually better to remove the underlying cause of
infeasibility at the modelling stage.
Indeed consider the example
minimize
subject to x 1 + x 2 = 1,
x 3 + x 4 = 1,
− x 1 − x 3 = − 1 + ɛ
− x 2 − x 4 = − 1,
x 1 , x 2 , x 3 , x 4 ≥ 0
(14.1)
then if we add the equalties together we obtain the implied equality
0 = ɛ
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