The MOSEK command line tool Version 7.0 (Revision 141)

4.5. GENERAL CONVEX OPTIMIZATION 29
The dual problem is related to the dual problem for linear optimization (see Section 4.2), 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 (4.14) and the dual problem (4.15).
4.4.2 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 4.1.2).
4.4.2.1 Primal infeasible problems
If the problem (4.14) 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 4.1.2.1).
4.4.2.2 Dual infeasible problems
If the problem (4.15) 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 ,
(4.16)
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,
4.5 General convex optimization
MOSEK is capable of solving smooth (twice differentiable) convex nonlinear optimization problems of
the form