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

98 CHAPTER 8. A CASE STUDY
Furthermore, if f ∈ R m×n then
⎡
f(:) =
⎢
⎣
f 1,1
f 2,1
f m−1,n
⎤
⎥
⎦
f m,n
i.e. f(:) stacks the columns of the matrix f.
8.1.2 A basic portfolio optimization model
The classical Markowitz portfolio optimization problem considers investing in n stocks or assets held
over a period of time. Let x j denote the amount invested in asset j, and assume a stochastic model
where the return of the assets is a random variable r with known mean
and covariance
µ = Er
Σ = E(r − µ)(r − µ) T .
The return of the investment is also a random variable y = r T x with mean (or expected return)
and variance (or risk)
Ey = µ T x
(y − Ey) 2 = x T Σx.
The problem facing the investor is to rebalance the portfolio to achieve a good compromise between
risk and expected return, e.g., maximize the expected return subject to a budget constraint and an
upper bound (denoted γ) on the tolerable risk. This leads to the optimization problem
maximize µ T x
subject to e T x = w + e T x 0 ,
x T Σx ≤ γ 2 ,
x ≥ 0.
(8.1)
The variables x denote the investment i.e. x j is the amount invested in asset j and x 0 j is the initial
holding of asset j. Finally, w is the initial amount of cash available.
A popular choice is x 0 = 0 and w = 1 because then x j may be interpretated as the relative amount of
the total portfolio that is invested in asset j.
Since e is the vector of all ones then