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

10.4. QUADRATIC AND QUADRATICALLY CONSTRAINED OPTIMIZATION 133
The dual problem is related to the dual problem for linear optimization (see Section 10.2), but depend
on variable x which in general can not be eliminated. In the solutions reported by MOSEK, the value
of x is the same for the primal problem (10.14) and the dual problem (10.15).
10.4.2 Infeasibility for quadratic and quadratically constrained 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).
10.4.2.1 Primal infeasible problems
If the problem (10.14) with all Q k = 0 is infeasible, MOSEK will report a certificate of primal infeasibility.
As the constraints is the same as for a linear problem, the certificate of infeasibility is the same
as for linear optimization (see Section 10.1.2.1).
10.4.2.2 Dual infeasible problems
If the problem (10.15) with all Q k = 0 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
where
minimize
subject to
c T x
ˆlc ≤ Ax ≤ û c ,
0 ≤ Q o x ≤ 0,
ˆlx ≤ x ≤ û x ,
(10.16)
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,