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

112 CHAPTER 8. A CASE STUDY
c j ≥ z p/q
j
where p and q are both integers and p ≥ q can be modelled using quadratic cones. See [7] for details.
One more reformulation of (8.13) is needed,
maximize µ T x
subject to e T x + m T c = w + e T x 0 ,
G T x − t = 0,
z j − x j ≥ − x 0 j, j = 1, . . . , n,
z j + x j ≥ x 0 j, j = 1, . . . , n,
[v j ; c j ; z j ] − f j,1:3 = 0, j = 1, . . . , n,
[z j ; 0; v j ] − g j,1:3 = [0; −1/8; 0], j = 1, . . . , n,
[s; t] ∈ Q n+1 ,
fj,1:3 T ∈ Q 3 r, j = 1, . . . , n,
gj,1:3 T ∈ Q 3 r, j = 1, . . . , n,
x ≥ 0,
s = γ,
(8.14)
where f, g ∈ R n×3 . These additional variables f and g are only introduced to bring the problem on
the API standard form.
The formulation (8.14) is not the most compact possible. However, the MOSEK presolve will automatically
make it more compact and since it is easier to implement (8.14) than a more compact form
then the form (8.14) is preferred.
The first step in developing the optimizer API implementation is to chose an ordering of the variables.
In this case the ordering
⎡
ˆx =
⎢
⎣
x
s
t
c
v
z
f T (:)
g T (:)
will be used. Note f T (:) means the rows of f are transposed and stacked on top of each other to form
a long column vector. The Table 8.2 shows the mapping between the ˆx and the model variables.
⎤
⎥
⎦
The next step is to consider how the columns of A is defined. Reusing the idea in Section 8.1.2 then
the following pseudo code describes the setup of A.