Portfolio Optimization sum of weights constraint with short selling

For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint.

In most resources I've seen, such as https://www.coursera.org/learn/financial-engineering-2/lecture/qwIYs/overview-of-mean-variance (week 1 first video), it is stated as:

$$\sum_{i=1}^N x_i = 1$$

However, in Tucker Balch's course https://classroom.udacity.com/courses/ud501/lessons/4432279076/concepts/44338591400923 (Lesson 02-04, Lecture 2), it is stated as:

$$\sum_{i=1}^N |x_i| = 1$$

Which one is correct? What is the reasoning behind it?

• Could you provide a bit more context on the second equation? The link you provided needs an account to view. Aug 2, 2020 at 9:42

You are free to choose whichever assumption you think is more appropriate for your situation (depending on local regulations and your broker's policy), or even to prohibit shorting entirely by requiring all $$x_i \ge 0$$ (in which case taking absolute values or not does not matter anymore). If you have no opinion and just want my recommendation I would say: use the first method. FWIW (and I don't want to advertise a specific firm) my account at Inter$$**$$tive Brokers allows me to take short positions corresponding to the first equation, and I am certainly not a big institutional investor. The only reason I see for teaching the second approach is if you want to stay consistent with Markowitz's original paper.
• There are managers who use the second form -- or the equivalent $\sum_i^N x_i = 1$ with $x_i\geq0~\forall i=\{1,\ldots,N\}$. Mutual funds, for example, are often prohibited from any short selling. The growth in "130-30" funds a decade ago was a reaction to concerns that no short selling unduly hurt performance. Aug 3, 2020 at 3:50