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

110 CHAPTER 8. A CASE STUDY
where the function
maximize
µ T x
subject to e T x +
n∑
C j (x j − x 0 j) = w + e T x 0 ,
j=1
x T Σx ≤ γ 2 ,
x ≥ 0,
(8.8)
C j (x j − x 0 j)
specifies the transaction costs when the holding of asset j is changed from its initial value.
8.1.5.1 Market impact costs
If the initial wealth is fairly small and short selling is not allowed, then the holdings will be small.
Therefore, the amount traded of each asset must also be small. Hence, it is reasonable to assume that
the prices of the assets is independent of the amount traded. However, if a large volume of an assert is
sold or purchased it can be expected that the price change and hence the expected return also change.
This effect is called market impact costs. It is common to assume that market impact costs for asset
j can be modelled by
m j
√|x j − x 0 j |
where m j is a constant that is estimated in some way. See [6][p. 452] for details. To summarize then
From [7] it is known
√
C j (x j − x 0 j) = m j |x j − x 0 j| |x j − x 0 j | = m j|x j − x 0 j| 3/2 .
{(c, z) : c ≥ z 3/2 , z ≥ 0} = {(c, z) : [v; c; z], [z; 1/8; v] ∈ Q 3 r}
where Q 3 r is the 3 dimensional rotated quadratic cone implying
z j = |x j − x 0 j|,
[v j ; c j ; z j ], [z j ; 1/8; v j ] ∈ Q 3 r,
n∑
n∑
C j (x j − x 0 j) = c j .
j=1
j=1
Unfortunately this set of constraints is nonconvex due to the constraint
z j = |x j − x 0 j| (8.9)
but in many cases that constraint can safely be replaced by the relaxed constraint