The MOSEK command line tool Version 7.0 (Revision 141)

22 CHAPTER 4. PROBLEM FORMULATION AND SOLUTIONS
are satisfied.
(s c l ) ∗ i ((x c i) ∗ − li c ) = 0, i = 0, . . . , m − 1,
(s c u) ∗ i (u c i − (x c i) ∗ ) = 0, i = 0, . . . , m − 1,
(s x l ) ∗ j (x ∗ j − lj x ) = 0, j = 0, . . . , n − 1,
(s x u) ∗ j (u x j − x ∗ j ) = 0, j = 0, . . . , n − 1,
If (4.1) has an optimal solution and MOSEK solves the problem successfully, both the primal and dual
solution are reported, including a status indicating the exact state of the solution.
4.1.2 Infeasibility for linear optimization
4.1.2.1 Primal infeasible problems
If the problem (4.1) is infeasible (has no feasible solution), MOSEK will report a certificate of primal
infeasibility: The dual solution reported is the certificate of infeasibility, and the primal solution is
undefined.
A certificate of primal infeasibility is a feasible solution to the modified dual problem
maximize (l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u
subject to A T y + s x l − s x u = 0,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0,
such that the objective value is strictly positive, i.e. a solution
(4.4)
to (4.4) so that
(y ∗ , (s c l ) ∗ , (s c u) ∗ , (s x l ) ∗ , (s x u) ∗ )
(l c ) T (s c l ) ∗ − (u c ) T (s c u) ∗ + (l x ) T (s x l ) ∗ − (u x ) T (s x u) ∗ > 0.
Such a solution implies that (4.4) is unbounded, and that its dual is infeasible. As the constraints to
the dual of (4.4) is identical to the constraints of problem (4.1), we thus have that problem (4.1) is
also infeasible.
4.1.2.2 Dual infeasible problems
If the problem (4.2) is infeasible (has no feasible solution), MOSEK will report a certificate of dual
infeasibility: The primal solution reported is the certificate of infeasibility, and the dual solution is
undefined.
A certificate of dual infeasibility is a feasible solution to the modified primal problem
minimize
subject to
ˆlc
ˆlx
c T x
≤ Ax ≤ û c ,
≤ x ≤ û x ,
(4.5)