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

8.1. PORTFOLIO OPTIMIZATION 109
110 print("An error occurred when solving for alpha=%e\n" % alpha)
8.1.4 Improving the computational efficiency
In practice it is often important to solve the portfolio problem in a short amount of time; this section
it is discusses what can be done at the modelling stage to improve the computational efficiency.
The computational cost is of course to some extent dependent on the number of constraints and
variables in the optimization problem. However, in practice a more important factor is the number
nonzeros used to represent the problem. Indeed it is often better to focus at the number of nonzeros
in G (see (8.2)) and try to reduce that number by for instance changing the choice of G.
In other words, if the computational efficiency should be improved then it is always good idea to start
with focusing at the covariance matrix. As an example assume that
Σ = D + V V T
where D is positive definite diagonal matrix. Moreover, V is a matrix with n rows and p columns.
Such a model for the covariance matrix is called a factor model and usually p is much smaller than n.
In practice p tends be a small number say less than 100 independent of n.
One possible choice for G is the Cholesky factorization of Σ which requires storage proportional to
n(n + 1)/2. However, another choice is
because then
G T =
[
D
1/2
V T ]
GG T = D + V V T .
This choice requires storage proportional to n + pn which is much less than for the Cholesky choice of
G. Indeed assuming p is a constant then the difference in storage requirements is a factor of n.
The example above exploits the so-called factor structure and demonstrates that an alternative choice
of G may lead to a significant reduction in the amount of storage used to represent the problem. This
will in most cases also lead to a significant reduction in the solution time.
The lesson to be learned is that it is important to investigate how the covariance is formed. Given
this knowledge it might be possible to make a special choice for G that helps reducing the storage
requirements and enhance the computational efficiency.
8.1.5 Slippage cost
The basic Markowitz portfolio model assumes that there are no costs associated with trading the assets
and that the returns of the assets is independent of the amount traded. None of those assumptions
are usually valid in practice. Therefore, a more realistic model is