Consider the following optimization problem: \begin{equation*} \begin{aligned} & \underset{z\in\mathbb{R},\, v\in\mathbb{R}^{d}}{\text{maximize}} & & z \\ & \text{subject to} & & z \geq \varepsilon,\\ & & & v \geq \varepsilon\mathbb{1},\\ & & & (zI+A)v \leq 0, \end{aligned} \end{equation*} for some (small) $\varepsilon>0$ and Metzler matrix $A\in\mathbb{R}^{d\times d}$.
Note that there is a bilinear constraint in the optimization problem above. Can this problem be rephrased as a standard convex optimization problem, e.g. semidefinite programming, and simply solved by existing software?