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

174 CHAPTER 13. THE ANALYZERS
13.2.5 Theory concerning infeasible problems
This section discusses the theory of infeasibility certificates and how MOSEK uses a certificate to
produce an infeasibility report. In general, MOSEK solves the problem
where the corresponding dual problem is
(13.3)
minimize
c T x + c f
subject to l c ≤ Ax ≤ u c ,
l x ≤ x ≤ u x
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 + c f
subject to A T y + s x l − s x u = c,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0.
(13.4)
We use the convension that for any bound that is not finite, the corresponding dual variable is fixed
at zero (and thus will have no influence on the dual problem). For example
l x j = −∞ ⇒ (s x l ) j = 0
13.2.6 The certificate of primal infeasibility
A certificate of primal infeasibility is any solution to the homogenized dual problem
with a positive objective value. That is, (s c∗
l
, s c∗
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.
u , s x∗
l
, s x∗
u ) is a certificate of primal infeasibility if
and
(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
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.
The well-known Farkas Lemma tells us that (13.3) is infeasible if and only if a certificate of primal
infeasibility exists.
Let (s c∗
l
, s c∗
u , s x∗
l
, s x∗
u ) be a certificate of primal infeasibility then