# How is the inequality constraint $\mbox{Tr}(W) \geq c$ convex?

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?

• 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! – Ariel Serranoni May 21 at 13:56

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.

• Thank you for the answer. – Frank Moses May 21 at 14:28

$$\text{Tr}(W)$$ is a linear function, so the constraint is convex.