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I am trying to make the following code work:

M = {{x11, x12, x13}, {x21, x22, x23}, {x31, x32, x33}};
SemidefiniteOptimization[
  1, 
  VectorGreaterEqual[{M, 0}, {"SemidefiniteCone", 3}], 
  Flatten[M]]

However, it just stops and does nothing. Essentially, the SDP problem would consist on finding the first matrix $M$ that is positive semidefinite.

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    $\begingroup$ Does VectorGreaterEqual[{M, 0} impliy semidefiniteness of M? $\endgroup$ – user64494 Oct 12 '20 at 13:08
  • $\begingroup$ I think it does, at least that is the translation of the symbol appearing in the reference. reference.wolfram.com/language/ref/… $\endgroup$ – bdt Oct 12 '20 at 13:22
  • $\begingroup$ Indeed, this quits the kernel. Must be a memory issue. This must not happen. Please contact Wolfram Support about this. $\endgroup$ – Henrik Schumacher Oct 12 '20 at 13:53
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Up to the documentation,

The matrices Subscript[a, j] must be symmetric n *n matrices.

Therefore, M should be changed to become a symmetric matrix. The following works.

M = {{x11, x12, x13}, {x12, x22, x23}, {x13, x23, x33}};
SemidefiniteOptimization[1, VectorGreaterEqual[{M, 0}, {"SemidefiniteCone", 3}], {x11, x12, x13, 
x22, x23, x33}]
(*{x11 -> 20.9077, x12 -> 0., x13 -> 0., x22 -> 20.9077, x23 -> 0., x33 -> 20.9077}*)
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