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

10.2. CONIC QUADRATIC OPTIMIZATION 129
Please note that the dual problem of the dual problem is identical to the original primal problem.
10.2.2 Infeasibility for conic quadratic optimization
In case MOSEK finds a problem to be infeasible it reports a certificate of the infeasibility. This works
exactly as for linear problems (see Section 10.1.2).
10.2.2.1 Primal infeasible problems
If the problem (10.6) is infeasible, 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 problem
such that the objective value is strictly positive.
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 + s x n = 0,
− y + s c l − s c u = 0,
s c l , s c u, s x l , s x u ≥ 0,
s x n ∈ C ∗ ,
(10.8)
10.2.2.2 Dual infeasible problems
If the problem (10.7) is infeasible, 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 problem
minimize
subject to
ˆlc
ˆlx
c T x
≤ Ax ≤ û c ,
≤ x ≤ û x ,
x ∈ C,
(10.9)
where
and
{
0 if l
c
ˆlc i = i > −∞,
− ∞ otherwise,
{
and û c 0 if u
c
i :=
i < ∞,
∞ otherwise,
{ 0 if l
x
ˆlx j = j > −∞,
− ∞ otherwise,
such that the objective value is strictly negative.
{ 0 if u
and û x x
j :=
j < ∞,
∞ otherwise,