# Closed form solution for Mean-Variance optimization without short-selling

So I am writing my bachelor thesis about the naive portfolio vs mean-variance portfolio and I am currently a bit stuck at the part about describing the mean-variance portfolio. I know that if there are only constraints in the form of equalities, you can use the Lagrangian method in order to find a closed form solution for the problem: $$min\frac{\lambda}{2}w\Sigma w^T-w^T\mu$$ But in my problem I have the following constraints: $$e^Tw=1$$ and $$0\leq w_i\leq1$$, with e being the vectors of ones. Can I only find a optimal weight allocation by using quadratic programming?

• Because of the no short selling constraint the answer is unfortunately yes, there is no analytic solution for this case. It requires applying an optimization algorithm. Commented Oct 29, 2023 at 12:24

$$$$w_{min} = \frac{\Sigma^{-1} e}{e^T \Sigma^{-1} e}$$$$
with $$e=[1, 1, \cdots, 1]$$. The only difference being that you can short securities, so you no longer have the purely positive weights constraints. However, I am more of a fan of the quadratic programming approach. Hopefully this gives you some ideas?