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

8.1. PORTFOLIO OPTIMIZATION 99
e T x =
is the total investment. Clearly, the total amount invested must be equal to the initial wealth, which
is
n∑
j=1
x j
This leads to the first constraint
w + e T x 0 .
The second constraint
e T x = w + e T x 0 .
x T Σx ≤ γ 2
ensures that the variance, or the risk, is bounded by γ 2 . Therefore, γ specifies an upper bound of the
standard deviation the investor is willing to undertake. Finally, the constraint
x j ≥ 0
excludes the possibility of short-selling. This constraint can of course be excluded if short-selling is
allowed.
The covariance matrix Σ is positive semidefinite by definition and therefore there exist a matrix G
such that
Σ = GG T . (8.2)
In general the choice of G is not unique and one possible choice of G is the Cholesky factorization of
Σ. However, in many cases another choice is better for efficiency reasons as discussed in Section 8.1.4.
For a given G we have that
Hence, we may write the risk constraint as
x T Σx = x T GG T x
= ∥ ∥ G T x ∥ ∥ 2 .
or equivalently
γ ≥ ∥ ∥ G T x ∥ ∥
[γ; G T x] ∈ Q n+1 .
where Q n+1 is the n + 1 dimensional quadratic cone. Therefore, problem (8.1) can be written as