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

10.2. CONIC QUADRATIC OPTIMIZATION 127
where
{ 0 if l
c
ˆlc i = i > −∞,
− ∞ otherwise,
{
and û c 0 if u
c
i :=
i < ∞,
∞ otherwise,
and
{ 0 if l
x
ˆlx j = j > −∞,
− ∞ otherwise,
{ 0 if u
and û x x
j :=
j < ∞,
∞ otherwise,
such that the objective value c T x is strictly negative.
Such a solution implies that (10.5) is unbounded, and that its dual is infeasible. As the constraints to
the dual of (10.5) is identical to the constraints of problem (10.2), we thus have that problem (10.2) is
also infeasible.
10.1.2.3 Primal and dual infeasible case
In case that both the primal problem (10.1) and the dual problem (10.2) are infeasible, MOSEK will
report only one of the two possible certificates — which one is not defined (MOSEK returns the first
certificate found).
10.2 Conic quadratic optimization
Conic quadratic optimization is an extensions of linear optimization (see Section 10.1) allowing conic
domains to be specified for subsets of the problem variables. A conic quadratic optimization problem
can be written as
minimize
c T x + c f
subject to l c ≤ Ax ≤ u c ,
l x ≤ x ≤ u x ,
x ∈ C,
where set C is a Cartesian product of convex cones, namely C = C 1 × · · · ×C p .
restriction, x ∈ C, is thus equivalent to
(10.6)
Having the domain
x t ∈ C t ⊆ R nt ,
where x = (x 1 , . . . , x p ) is a partition of the problem variables. Please note that the n-dimensional
Euclidean space R n is a cone itself, so simple linear variables are still allowed.
MOSEK supports only a limited number of cones, specifically:
• The R n set.