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

8.1. PORTFOLIO OPTIMIZATION 111
z j ≥ |x j − x 0 j| (8.10)
which is convex. If for instance the universe of assets contains a risk free asset with a positive return
then
z j > |x j − x 0 j| (8.11)
cannot hold for an optimal solution because that would imply the solution is not optimal.
Now assume that the optimal solution has the property that (8.11) holds then the market impact cost
within the model is larger than the true market impact cost and hence money are essentially considered
garbage and removed by generating transaction costs. This may happen if a portfolio with very small
risk is requested because then the only way to obtain a small risk is to get rid of some of the assets by
generating transaction costs. Here it is assumed this is not the case and hence the models (8.9) and
(8.10) are equivalent.
Formula (8.10) is replaced by constraints
z j ≥ x j − x 0 j,
z j ≥ − (x j − x 0 j).
(8.12)
Now we have
The revised budget constraint
maximize µ T x
subject to e T x + m T c = w + e T x 0 ,
z j ≥ x j − x 0 j, j = 1, . . . , n,
z j ≥ x 0 j − x j , j = 1, . . . , n,
[γ; G T x] ∈ Q n+1 ,
[v j ; c j ; z j ] ∈ Q 3 r, j = 1, . . . , n,
[z j ; 1/8; v j ] ∈ Q 3 r, j = 1, . . . , n,
x ≥ 0.
(8.13)
e T x = w + e T x 0 − m T c
specifies that the total investment must be equal to the initial wealth minus the transaction costs.
Moreover, observe the variables v and z are some auxiliary variables that model the market impact
cost. Indeed it holds
and
z j ≥ |x j − x 0 j|
c j ≥ z 3/2
j .
Before proceeding it should be mentioned that transaction costs of the form