I'm using CVXPY. Given a $2\times2$ matrix $A$, is it possible to add a singularity constraint?

Anything equivalent to:

  • $|A| = 0$: The determinant is $0$
  • $\operatorname{rank}(A) \leq 1$
  • Smallest eigenvalue is $0$
  • etc...
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    $\begingroup$ X-posted. I would expect someone with some 15K rep points on SO to know the etiquette. $\endgroup$ – Rodrigo de Azevedo Mar 26 at 12:43
  • $\begingroup$ @RodrigodeAzevedo Yes, I'm aware - it's just that this site is a better fit but still in Beta so I wasn't sure about the right place to put it... I should have at least put a cross-link. Sorry. $\endgroup$ – Adi Shavit Mar 26 at 12:45
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    $\begingroup$ Try to use the nuclear norm instead of the rank. It is convex, at least. $\endgroup$ – Rodrigo de Azevedo Mar 26 at 13:50
  • $\begingroup$ With proper scaling this does seem to do the trick when I put it in the objective and minimize it. It does not let me create a constraint with it though. $\endgroup$ – Adi Shavit Mar 26 at 14:48
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    $\begingroup$ I don't have the big picture. If you edited your question and showed all your work and made an effort to present it clearly, it would be easier for everyone. $\endgroup$ – Rodrigo de Azevedo Mar 26 at 15:15

No, it is an intrinsically non-convex constraint. Just take a diagonal matrix, and the feasible set would be the coordinate axes, i.e. nonconvex and highly ill-conditioned as the feasible set has measure 0.

  • $\begingroup$ I have an upper bound and a lower for all the matrix values. Does that help? $\endgroup$ – Adi Shavit Mar 26 at 11:05
  • $\begingroup$ No, you're still looking at the coordinate axes in the special case, but now constrained to a box. In general for a 2x2 matrix you have a quadratic equality $a_1 a_4-a_2 a_3 = 0$ $\endgroup$ – Johan Löfberg Mar 26 at 11:09
  • $\begingroup$ Yes. Ofc. But that’s non affine and CVXPY won’t let me add that expression. It’s non DCP. $\endgroup$ – Adi Shavit Mar 26 at 11:11
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    $\begingroup$ Yes, that is precisely what this answer says. It is a non-convex constraint. $\endgroup$ – Johan Löfberg Mar 26 at 11:21

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