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So what I am trying to do is to minimize the distance between the Frobenius norm of a PSD matrix and a real positive value, which can be formulated as $$\min \left|\|\textbf{P}\|_F - J\right|^2$$ where $\textbf{P}$ is the PSD matrix to be optimized and $J$ is a real positive constant. Then the problem here is that the whole minimization will fail the disciplined convex programming rule, and I want to know whether we can formulate this idea to be a convex formulation and accepted by CVX. Is it possible to do that?


Edit: $\textbf{P}\in \mathbb{C}^{J \times J}$.

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    $\begingroup$ Can you diagonalize the matrix beforehand? Then you can just work on the eigenvalues with a positivity constraint and massively reduce your dimensionality $\endgroup$
    – whpowell96
    Commented Nov 21, 2023 at 16:31
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    $\begingroup$ You don't have a convex objective function being minimized, so it won't be possible to formulate this in DCP. $\endgroup$ Commented Nov 21, 2023 at 21:08
  • $\begingroup$ @whpowell96 Really thanks for the comment. It is a good option, but for my specific problem, the PSD matrix that to be optimized has nontrivial off-diagonal entries, so I don't think I can directly diagonalize here. $\endgroup$
    – tyrela
    Commented Nov 22, 2023 at 1:25
  • $\begingroup$ @BrianBorchers Really thanks for the comment. Yes, unfortunately it's a nonconvex problem here. I was just trying my luck, maybe there are some tricks to approximate this nonconvex problem to convex that I just don't know. Right now, it seems I should try some nonconvex solver. $\endgroup$
    – tyrela
    Commented Nov 22, 2023 at 1:30
  • $\begingroup$ What does it mean to "optimize" your matrix? You want to find the closest PSD matrix to some initial $P$ that satisfies $\|P\| = J$? It's not clear what properties an optimal solution to your problem should have $\endgroup$
    – whpowell96
    Commented Nov 22, 2023 at 14:46

1 Answer 1

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One solution to the problem is to choose $P$ as the diagonal matrix with diagonal entries equal to $J/\sqrt{n}$. Its Frobenius norm is $J$ and it is symmetric and positive definite.

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    $\begingroup$ I suppose the real question is what extra conditions $P$ must satisfy to make this nontrivial. Should it be minimally distant from the some initial $P_0$? Otherwise you can just always choose a trivial solution. $\endgroup$
    – whpowell96
    Commented Nov 21, 2023 at 17:25
  • $\begingroup$ Really thanks for the answer. Yes, that will be great, but for my specific problem, I have a PSD with nontrivial offdiagonal entries. So, still thinking whether I can approximate this problem to be convex and solve it. $\endgroup$
    – tyrela
    Commented Nov 22, 2023 at 1:41

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