# Mean variance portfolio optimization with long short positions

Sorry if this has been asked before. Can someone point me to some places explaining how to set up the mean variance optimization on long short portfolios? Classical formulation has long only.

Is it as simple as removing $$w_{i} >0$$ constraint but leave $$\sum{w_{i}} = 1$$ unchanged? I remember $$w_{i}$$ is dollar notional weight of positions i over NET value of portfolio p. Won't $$w_{i}$$ be undefined if the portfolio is a perfectly long short balanced portfolio?

Also, I read some URLs talking about splitting the holdings to positive and negative or duplicate return matrix. Kind of lost here. Can someone point me to some directions?

Thanks

• There are several ways to do it. (1) Simply remove the $w_i>0$ constraint. You will get a net long portfolio, with some short positions. (2) If you want a "perfectly long short balanced portfolio" (which is sometimes called an "arbitrage portfolio" in the literature) then in addition you should change the other constraint to $\sum w_i=0$. (Keep in mind that with short positions allowed you may find unrealistically large $|w_i|$ pop up in your solution. You may want to constrain weights to be in a reasonable range). Apr 22 at 12:19