The MOSEK command line tool Version 7.0 (Revision 141)

38 CHAPTER 5. THE OPTIMIZERS FOR CONTINUOUS PROBLEMS
A x∗
τ ∗ = b,
A T y∗
τ ∗ + s∗
= c,
τ∗
− c T x∗ y∗
+ bT
τ
∗
τ ∗ = 0,
x ∗ , s ∗ , τ ∗ , κ ∗ ≥ 0.
This shows that x∗
τ
is a primal optimal solution and ( y∗
∗
as the optimal interior-point solution since
is a primal-dual optimal solution.
On other hand, if κ ∗ > 0 then
(x, y, s) =
τ
, s∗
∗ τ∗
( )
x
∗
τ ∗ , y∗
τ ∗ , s∗
τ∗
) is a dual optimal solution; this is reported
This implies that at least one of
Ax ∗ = 0,
A T y ∗ + s ∗ = 0,
− c T x ∗ + b T y ∗ = κ ∗ ,
x ∗ , s ∗ , τ ∗ , κ ∗ ≥ 0.
or
− c T x ∗ > 0 (5.3)
b T y ∗ > 0 (5.4)
is satisfied. If (5.3) is satisfied then x ∗ is a certificate of dual infeasibility, whereas if (5.4) is satisfied
then y ∗ is a certificate of dual infeasibility.
In summary, by computing an appropriate solution to the homogeneous model, all information required
for a solution to the original problem is obtained. A solution to the homogeneous model can be
computed using a primal-dual interior-point algorithm [5].
5.2.2.1 Interior-point termination criterion
For efficiency reasons it is not practical to solve the homogeneous model exactly. Hence, an exact
optimal solution or an exact infeasibility certificate cannot be computed and a reasonable termination
criterion has to be employed.
In every iteration, k, of the interior-point algorithm a trial solution
to homogeneous model is generated where
(x k , y k , s k , τ k , κ k )