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

14.2. AUTOMATIC REPAIR 179
One way of making the problem feasible is to reduce the lower bounds and increase the upper bounds.
If the change is sufficiently large the problem becomes feasible. Now an obvious idea is to compute
the optimal relaxation by solving an optimization problem. The problem
minimize p(v c l , v c u, v x l , v x u)
subject to l c ≤ Ax + v c l − v c u ≤ u c ,
l x ≤ x + v x l − v x u ≤ u x ,
v c l , v c u, v x l , v x u ≥ 0
(14.4)
does exactly that. The additional variables (v c l ) i, (v c u) i , (v x l ) j and (v c u) j are elasticity variables because
they allow a constraint to be violated and hence add some elasticity to the problem. For instance,
the elasticity variable (v c l ) i controls how much the lower bound (l c ) i should be relaxed to make the
problem feasible. Finally, the so-called penalty function
p(vl c , vu, c vl x , vu)
x
is chosen so it penalize changes to bounds. Given the weights
• wl c ∈ Rm (associated with l c ),
• wu c ∈ R m (associated with u c ),
• wl x ∈ R n (associated with l x ),
• wu x ∈ R n (associated with u x ),
then a natural choice is
p(v c l , v c u, v x l , v x u) = (w c l ) T v c l + (w c u) T v c u + (w x l ) T v x l + (w x u) T v x u. (14.5)
Hence, the penalty function p() is a weighted sum of the relaxation and therefore the problem (14.4)
keeps the amount of relaxation at a minimum. Please observe that
• the problem (14.6) is always feasible.
• a negative weight implies problem (14.6) is unbounded. For this reason if the value of a weight
is negative MOSEK fixes the associated elasticity variable to zero. Clearly, if one or more of the
weights are negative may imply that it is not possible repair the problem.
A simple choice of weights is to let them all to be 1, but of course that does not take into account that
constraints may have different importance.
14.2.1 Caveats
Observe if the infeasible problem