Mean variance portfolio - alternative formulations

From this lecture on YouTube the lecturer states that there are three ways to form the mean variance portfolio (minimize variance for a given return, maximize return for a given variance, maximize a "risk adjusted return" which is the return minus a penalty for variance) however he only goes into the first in detail. Are there any texts which cover the formulations of problems two and three? Lecture notes which accompany the series are here and problems are on slide 7.

I have found that "Robust Portfolio Optimization and Management" mentions alternative versions, however doesn't include a derivation.

• You should be able to derive the the other two formulations using the same method i.e. Lagrange Multipliers. The conclusions are the same. Just 3 slightly different ways of doing it. – noob2 May 9 at 8:11
• Thank you for responding. My reason for posting is that I'm struggling with the math on the alternatives. For the method covered by the lecturer it follows a quadratic programming problem whereby I am minimizing subject to linear constraints. For the other two I am minimizing subject to a non-linear constraint. – SelfLearner May 9 at 8:31
• The 3 are (theoretically) equivalent, but you are right that the first is more convenient to solve on a computer because of the availability of codes for nonlinear optimization s.t. linear constraints. The second method is tough to solve numerically. The third is OK because again the constraint is linear (sum of weights = 1). So in applications you want to use Formulation 1 usually (or maybe 3 if you know the investor's risk aversion parameter $\lambda$). A common approach is to solve 1 iteratively for various values of $\alpha_0$ so you get multiple points along the whole efficient frontier. – noob2 May 9 at 14:09