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I am taking the convex optimization course by CMU (though I am not a CMU student), and got stuck on this problem.

Formally, show that computing $\left \| X \right \|_{tr}$ can be expressed as the following convex optimization problem: $$\begin{array}{ll} \underset{{Y \in \mathbb{R}^{m \times n}}}{\text{maximize}} & \mbox{tr} \left( X^T Y \right)\\ \text{subject to} & \begin{pmatrix} I_{m} & Y \\ Y^{T} & I_{n} \end{pmatrix} \succeq 0\end{array}$$ where $I_{p}$ is the $p \times p$ identity matrix.

Any help would be appropriated.

Rodrigo de Azevedo
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zhu11
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1 Answers1

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What are the eigenvalues of $$\begin{pmatrix} I_{m} & Y \\ Y^{T} & I_{n} \end{pmatrix}?$$ Well, consider an eigenvector $[p, q].$ Then:

$$ p + Y q = \lambda p,$$ and

$$Y^{T}p + q = \lambda q.$$

We see that $$Y q = (\lambda - 1) p,$$ while $$Y^{T} p = (\lambda - 1) q.$$ Applying $Y$ to the second equation we see that $$YY^{T} p = (\lambda -1)^2.$$

So the block matrix is positive semidefinite if and only if $I\succeq YY^{T},$ which is exactly the condition in the question pointed out by Rodrigo.

Igor Rivin
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  • Thanks for your answer. I believe my block matrix is the same as that of the pointed question. However, I am a bit confused about your last equation. Did you mean $YY^{T}=(\lambda-1)^{2}\times I_{m}$? If that is the case, then $YY^{T}\preceq I_{m}$ implies $0\leqslant \lambda\leqslant 2$, which is not required by the problem description, am I wrong? – zhu11 Mar 20 '21 at 12:51