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

5.6. THE SOLUTION SUMMARY 53
• The solution status is optimal. The solution status can be obtained using Task.getsolsta.
• Next information about the primal solution is reported. The information consists of the objective
value and violation meassures for the primal solution. In this case violations for the constraints
and variables are small meaning the solution is very close to being an exact feasible solution. The
violation meassure for the variables is the worst violation of the solution in any of the bounds
on the variables.
The constraint and variable violations are computed with Task.getpviolcon and Task.getpviolvar.
• Similarly for the dual solution the violations are small and hence the dual solution is feasible. The
constraint and variable violations are computed with Task.getdviolcon and Task.getdviolvar
respectively.
• Finally, it can be seen that the primal and dual objective values are almost identical. Using
Task.getprimalobj and Task.getdualobj the primal and dual objective values can be obtained.
To summarize in this case a primal and a dual solution with small feasiblity violations are available.
Moreover, the primal and dual objective values are almost identical and hence it can be concluded
that the reported solution is a good approximation to the optimal solution.
Now what happens if the problem does not have an optimal solution e.g. it is primal infeasible. In
that case the solution summary may look like
Basic solution summary
Problem status : PRIMAL INFEASIBLE
Solution status : PRIMAL INFEASIBLE CER
Dual. obj: 3.5894503823e+004 Viol. con: 0e+000 var: 2e-008
i.e. MOSEK reports that the solution is a certificate of primal infeasibility. Since the problem is primal
infeasible it does not make sense to report any information about the primal solution. However, the
dual solution should be a certificate of the primal infeasibility. If the problem is a minimization problem
then the dual objective value should be positive and in the case of a maximization problem it should be
negative. The quality of the certificate can be evaluated by comparing the dual objective value to the
violations. Indeed if the objective value is large compared to the largest violation then the certificate
highly accurate. Here is an example
Basic solution summary
Problem status : PRIMAL INFEASIBLE
Solution status : PRIMAL INFEASIBLE CER
Dual. obj: 3.0056574100e-005 Viol. con: 9e-013 var: 2e-011
of a not so strong infeasibility certificate because the dual objective value is small compared to largest
violation.
In the case a problem is dual infeasible then the solution summary may look like
Basic solution summary
Problem status : DUAL INFEASIBLE
Solution status : DUAL INFEASIBLE CER
Primal. obj: -1.4500853392e+001 Viol. con: 0e+000 var: 0e+000
Observe when a solution is a certificate of dual infeasibility then the primal solution contains the
certificate. Moreoever, given the problem is a minimization problem the objective value should negative
and the objective should be large compared to the worst violation if the certificate is strong.