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In one paper, I found that the following inequality constraint $$\mbox{Tr}(W) \geq c$$ where $W$ is a symmetric positive semidefinite matrix variable and $c$ is a constant, is convex.

In my understanding $\mbox{Tr}(W)$ is a convex function and, therefore, the above constraint should not define a convex set. Is there something wrong with my understanding?

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    $\begingroup$ Maybe I get your problem. Did you think that the epigraph of a convex function is convex and since you are dealing with the hipograph then it should be concave? If so, note that the trace is a linear function. In this case 'both sides' of its graph are convex! $\endgroup$ – Ariel Serranoni May 21 at 13:56
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Suppose $$Tr(X) \ge c$$

and $$Tr(Y) \ge c$$

For $\alpha \in (0,1)$, we have $$Tr(\alpha X + (1-\alpha) Y)=\alpha Tr(X)+(1-\alpha)Tr(Y) \ge c$$

Hence it is convex.

Note that trace is a linear function.

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  • $\begingroup$ Thank you for the answer. $\endgroup$ – Frank Moses May 21 at 14:28
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$\text{Tr}(W)$ is a linear function, so the constraint is convex.

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