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

128 CHAPTER 10. PROBLEM FORMULATION AND SOLUTIONS
• The quadratic cone:
⎧
⎨
Q n =
⎩ x ∈ n∑
Rn : x 1 ≥ √
j=2
x 2 j
⎫
⎬
⎭ .
• The rotated quadratic cone:
⎧
⎨
Q r n =
⎩ x ∈ Rn : 2x 1 x 2 ≥
⎫
n∑
⎬
x 2 j, x 1 ≥ 0, x 2 ≥ 0
⎭ .
j=3
Although these cones may seem to provide only limited expressive power they can be used to model a
wide range of problems as demonstrated in [7].
10.2.1 Duality for conic quadratic optimization
The dual problem corresponding to the conic quadratic optimization problem (10.6) is given by
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 + s x n = c,
− 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 ∗ ,
where the dual cone C ∗ is a Cartesian product of the cones
(10.7)
C ∗ = C ∗ 1× · · · ×C ∗ p,
where each C ∗ t is the dual cone of C t . For the cone types MOSEK can handle, the relation between the
primal and dual cone is given as follows:
• The R n set:
• The quadratic cone:
C t = R nt ⇔ C ∗ t = {s ∈ R nt : s = 0} .
⎧
⎨
C t = Q nt ⇔ Ct ∗ = Q nt =
⎩ s ∈ ∑n t
Rnt : s 1 ≥ √
j=2
s 2 j
⎫
⎬
⎭ .
• The rotated quadratic cone:
⎧
⎨
C t = Q r n t
⇔ Ct ∗ = Q r n t
=
⎩ s ∈ Rnt : 2s 1 s 2 ≥
∑n t
j=3
⎫
⎬
s 2 j, s 1 ≥ 0, s 2 ≥ 0
⎭ .