Mean Variance portfolio selection with l1-norm

I have to find several efficient borders, applying to the markovitz model an l1-norm, so that the sum of the weights within the portfolio gives 1 and the sum of the weights in absolute value is less than or equal to a certain t-value. So I have to find the vector between t-min and t-max in such a way that the first one corresponds to a portfolio composed by only one asset with weight 1, and the second one to the portfolio uncostrained. Can I do this optimization through Quadprog?

Answer 1

[Matt J]

Sure. Minimizing a quadratic subject to linear constraints is exactly what quadprog is for.

[ANDREA MUZI]

I'm trying to do it, but I don't know how to write the variable u

[Matt J]

Ah, wait. Your first constraint is nonlinear in x and u, so no, you cannot use quadprog for this. You will have to use fmincon. As for how to handle a case where you have two variables x and u, you must combine them into a single concatenated vector z=[x;u] and write everything in terms of the single unknown variable z.

[ANDREA MUZI]

I think that the problem is only on the variable u. I did other optimizations with the first constraint using quadprog. Deleting the first constraint, I can find the vector of weights that minimize the variance and so multiplying this vector for the mean of the returns I obtain the minimum expected return portfolio

[Matt J]

If you delete the first constraint, the problem as you've written it is solvable with quadprog in both x and u. However, with the first constraint included, and with both x and u unknown, it is no longer in the scope of quadprog.