The MOSEK command line tool Version 7.0 (Revision 141)

7.2. ANALYZING INFEASIBLE PROBLEMS 69
(s c∗
l ) i > 0((s c∗
u ) i > 0)
implies that the lower (upper) bound on the i th constraint is important for the infeasibility. Furthermore,
(s x∗
l ) j > 0((s x∗
u ) i > 0)
implies that the lower (upper) bound on the j th variable is important for the infeasibility.
7.2.7 The certificate of dual infeasibility
A certificate of dual infeasibility is any solution to the problem
minimize
c T x
subject to ¯lc ≤ Ax ≤ ū c ,
¯lx ≤ x ≤ ū x
with negative objective value, where we use the definitions
and
{ 0, l
c
¯lc i := i > −∞,
− ∞, otherwise,
, ū c i :=
{ 0, u
c
i < ∞,
∞, otherwise,
{ 0, l
x
¯lx i := i > −∞,
− ∞, otherwise,
and ū x i :=
{ 0, u
x
i < ∞,
∞, otherwise.
Stated differently, a certificate of dual infeasibility is any x ∗ such that
¯lc
¯lx
c T x ∗ < 0,
≤ Ax ∗ ≤ ū c ,
(7.5)
≤ x ∗ ≤ ū x
The well-known Farkas Lemma tells us that (7.4) is infeasible if and only if a certificate of dual
infeasibility exists.
Note that if x ∗ is a certificate of dual infeasibility then for any j such that
variable j is involved in the dual infeasibility.
x ∗ j ≠ 0,