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I see that they have the same objective function and equality constraint $Ax = b$, but I am having trouble with showing that $Gx \preceq h $ can be written as $$x_1F_1 + \dots + x_nF_n + K \preceq 0$$ where $F_i, K \in S^n$ are symmetric matrices. I have been told that

If the matricies $F_i, K$ are diagonal, then $$x_1F_1 + \dots + x_nF_n + K$$ reduces to a set of linear inequalities.

but I don't see how this is. I see how the matrix $x_1F_1 + \dots + x_nF_n + K $ reduces to a set of linear systems $x_1F_{1i} + \dots + x_nF_{ni} + K_i$ for $i = 1, \dots, n$ (where $F_{ki}$ is diagonal matrix $F_k$'s $i$-th diagonal entry), but the inequality symbol is still in terms of positive-semi definiteness $\preceq 0 $.

Does the semi-definite inequality symbol just become a component-wise inequality for vectors? If so, can someone show how this is?

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    $\begingroup$ Your question is very unclear. It appears that you're interested in what happens in SDP when the matrix variables are restricted to being diagonal. Is that correct? $\endgroup$ – Brian Borchers May 6 at 2:16
  • $\begingroup$ Very related. $\endgroup$ – Rodrigo de Azevedo May 6 at 16:58

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