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

8.1. PORTFOLIO OPTIMIZATION 101
is nicer than
∥
∥G T x ∥ ∥ ≤ γ
x T Σx ≤ γ 2
for small and values of γ. For instance assume a γ of 10000 then γ 2 would 1.0e8 which introduces a
scaling issue in the model. Hence, using conic formulation it is possible to work with the standard
deviation instead of the variance, which usually gives rise to a better scaled model.
8.1.2.2 Implementing the portfolio model
The model (8.3) can not be implemented as stated using the MOSEK optimizer API because the API
requires the problem to be on the form
where ˆx is referred to as the API variable.
maximize
c T ˆx
subject to l c ≤ Aˆx ≤ u c ,
l x ≤ ˆx ≤ u x ,
ˆx ∈ K.
The first step in bringing (8.3) to the form (8.4) is the reformulation
(8.4)
maximize µ T x
subject to e T x = w + e T x 0 ,
G T x − t = 0
[s; t] ∈ Q n+1 ,
x ≥ 0,
s 0.
where s is an additional scalar variable and t is a n dimensional vector variable. The next step is to
define a mapping of the variables
(8.5)
ˆx = [x; s; t] =
⎡
⎣ x s
t
⎤
⎦ . (8.6)
Hence, the API variable ˆx is concatenation of model variables x, s and t. In Table (8.1) the details of
the concatenation are specified. For instance it can be seen that
because the offset of the t variable is n + 2.
ˆx n+2 = t 1 .
Given the ordering of the variables specified by (8.6) the data should be defined as follows