Given $A\in\mathbb{C}^{n\times n}$, I want to find out if the dual of the following problem is always solvable:
\begin{array}{ll} \underset{X\in\mathbb{C^{n\times n}}}{\text{minimize}} & \mathrm{trace}\big((A+A^*)X).\\ \text{subject to} & \mathrm{trace}(A^*AX)=1,\\&\mathrm{trace}(X)=1,\\&X\geq0.\end{array}
$L(X,\lambda_1,\lambda_2)=\mathrm{trace}\big((A+A^*)X)-\lambda_1(\mathrm{trace}(A^*AX)-1)-\lambda_2(\mathrm{trace}(X)-1)$, then
\begin{align} &\frac{\partial L(X,\lambda_1,\lambda_2)}{\partial X}=(A+A^*)-\lambda_1A^*A-\lambda_2I, \end{align}
and the corresponding dual problem is
\begin{array}{ll} \underset{\lambda_1,\lambda_2}{\text{maximize}} & \lambda_1+\lambda_2.\\ \text{subject to} & (A+A^*)-\lambda_1A^*A-\lambda_2I\succeq0.\end{array}
I am not sure if this is correct dual problem and how to show that it is not always solvable (i.e. not always feasible).