Sorry to ask a question about the basic Markowitz portfolio optimization. The example is from Mosek's example book.

The basic Markowitz portfolio optimization is formulated as:

The book mentioned we can simply remove the constraint $x \ge 0$ to allow short-sell in this formulation.

However, my question is about the conic formulation of original problem, which was given as:
enter image description here

Assuming we similarly remove the constraint $x \ge 0$ in this conic formulation, will it allow short-sell, i.e. $x \lt 0$? My feeling is yes since the cone $Q^{n+1}$ only requires $x_0$ which is $\gamma$ to be non-negative. I tried to change the sample codes below to remove the non-negativity constraint. enter image description here

But the results seems to be the same. Just would like to confirm if my understanding is correct.

Thank you!


1 Answer 1


Yes, of course you can remove the $x\ge0$ constraint if you like.

Maybe your data is such that that short selling is not worthwhile. Hard to say.

  • $\begingroup$ Removing the positive domain constraint breaks the budget constraint. $\endgroup$
    – Hunaphu
    Sep 3, 2021 at 12:55
  • $\begingroup$ Thanks all for answering. Master @ErlingMOSEK, thanks for your confirmation. 8 -) $\endgroup$
    – inf
    Sep 4, 2021 at 1:11

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