First, I would like to say that I am, to some extent, really familiar with Finance, but when it comes to optimization, I'm a little bit overwhelmed. I'm currently working on a research project where I need to impose RIDGE (Sum of Squared Terms - L2) and LASSO (Sum of Absolute Values - L1) as constraints to two portfolios: Minimum Variance and Sharpe Ratio. However, whilst Minimum Variance is easily computable by any NLP (or quadratic) solver (FYI, I'm using MATLAB), the Sharpe Ratio is a different story. My professor discouraged me from directly optimizing on the ratio.
How would one efficiently approach the optimization of the Sharpe Ratio: $\frac{R - rf}{\sigma}$. I have already merged $R - rf$ in a single variable which is the Excess Return, the resulting Ratio being: $\frac{R^+}{\sigma}$. The program would take this form:
${\underset{x}{\arg\min}}$ $\frac{ux}{\sqrt{x \Sigma x'}}$ s.t. $\sum w_i = 1$ where both $u$ (mean) and $\Sigma$ (var-covar matrix) are the first two moments estimated every period. Obviously, I am omitting the RIDGE and LASSO constraints here since they are a totally different question.
How would one approach this?
One variation that I often see (proposed by Sharpe itself (1978) if I am not mistaken) is the following:
${\underset{x}{\arg\min}}$ ${ux -x \Sigma x'}$ s.t. $\sum w_i = 1$.
Other constraints of the form $Aeq = b$ and $A \leq b$ can be implemented as well. It seems to me that we're simply substracting the variance instead of dividing by the standard deviation (I don't think using the variance or standard deviation really changes anything besides the scale of the ratio). Does this produce similar results to the ratio or is this a completely different program?
Thanks.