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

11.2. LINEAR OPTIMIZATION 141
Whenever the trial solution satisfies the criterion
yk
AT
∥
( ∣ ∣ (x k ) T s k ∣∣∣
min
(τ k ) 2 , c T x k
τ k − bT y k ) ∣∣∣
τ k
the interior-point optimizer is terminated and
x k , s k , τ k , κ k > 0.
∥ ∥∥∥ ∥ Axk τ k − b ∞ ≤ ɛ p (1 + ‖b‖ ∞ ),
∥
τ k + sk ∥∥∥
τ k − c ∞ ≤ ɛ d (1 + ‖c‖ ∞ ), and
(
≤
ɛ g max
1, min(∣ ∣c T x k∣ ∣ , ∣ ∣b T y k∣ )
∣ )
τ k ,
(11.5)
(x k , y k , s k )
τ k
is reported as the primal-dual optimal solution. The interpretation of (11.5) is that the optimizer is
terminated if
• xk
τ k
•
is approximately primal feasible,
( )
y
k
, sk
τ k τ k
is approximately dual feasible, and
• the duality gap is almost zero.
On the other hand, if the trial solution satisfies
−ɛ i c T x k >
‖c‖ ∞
∥ ∥ Ax
k ∞
max(1, ‖b‖ ∞ )
then the problem is declared dual infeasible and x k is reported as a certificate of dual infeasibility.
The motivation for this stopping criterion is as follows: First assume that ∥ ∥ Ax
k ∥ ∥ ∞ = 0 ; then x k is
an exact certificate of dual infeasibility. Next assume that this is not the case, i.e.
and define
∥ Ax
k ∥ ∥ ∞ > 0,
It is easy to verify that
¯x := ɛ i
max(1, ‖b‖ ∞ )
‖Ax k ‖ ∞ ‖c‖ ∞
x k .
‖A¯x‖ ∞ = ɛ i
max(1, ‖b‖ ∞ )
‖c‖ ∞
and − c T ¯x > 1,
which shows ¯x is an approximate certificate of dual infeasibility where ɛ i controls the quality of the
approximation. A smaller value means a better approximation.