I am working through this paper.
I want to implement the robust Bayesian optimization (see pages 6 onward) in Matlab using fmincon.
Here is a brief overview of my problem:
Let $\alpha$ be the vector of returns and let $w$ be ta set of weights representing a portfolio.
Then, our usual mean-variance problem is to find the resultant portfolio weights that minimize the portfolio variance $\sigma^2(x)=w' \Sigma w$.
In the robust framework, the vector of returns is assumed to lie in some uncertainty region, call it $U$. Here, the authors define that region to be the sphere centered at $\alpha$ with radius equal to $|\chi| \cdot \alpha$ with $\chi \in [0,1]$.
In other words, our optimization problem now becomes:
$\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$
where $\Sigma$ is a positive definite matrix and $\omega$ is a vector of weights that sum to 1.
The authors show that:
where $\phi$ is the angle between the two vectors $\alpha$ and $\omega$.$
They give the formula for the "the robust covariance matrix family" - (see page 7 eq 13) as well as the formula for the optimized weights (see page 7 eq 12).
I am trying to implement this in Matlab. However, one cannot use equations 12 or 13 as both equations require the optimized weights to be known.
My question is therefore, is there a method to implement a robust optimization such as this OR any suggestions as to how I could go about doing this???