I found this interesting SDP exercise problem on minimizing the second largest eigenvalue of
$$ A(\mathbf{x}) = A_0 + \sum_{i=1}^n x_i A_i $$
where $A_0, A_1, \dots, A_n$ are given real symmetric matrices. The given hint says this can be reformulated as an SDP with a rank-$1$ constraint.
I tried to solve this by following this similar thread. I first reformulated the original problem as follows
$$\begin{gathered} \max_{\mathbf{v}_1,\mathbf{v}_2,\mathbf{x}} \mathbf{v}_2^T A(\mathbf{x})\mathbf{v}_2\\ \mathbf{v}_1^T \mathbf{v}_1 = 1,\ \mathbf{v}_2^T \mathbf{v}_2 = 1,\ \mathbf{v}_1^T \mathbf{v}_2 = 0\\ \mathbf{v}_1^TA(\mathbf{x})\mathbf{v}_1\geq \mathbf{v}_2^TA(\mathbf{x})\mathbf{v}_2 \end{gathered}$$
using orthonormal vectors $\mathbf{v}_1, \mathbf{v}_2$. Then, I defined two rank-$1$ matrices
$$\mathbf{W}_1=\mathbf{v}_1\mathbf{v}_1^T$$
$$\mathbf{W}_2=\mathbf{v}_2\mathbf{v}_2^T$$
and get
$$\max_{\mathbf{W}_1,\mathbf{W}_2,\mathbf{x}} \text{ trace}(\mathbf{W}_2A(\mathbf{x}))\\ \mathbf{W}_1, \mathbf{W}_2\ \ \text{are rank one}\\ \text{trace}(\mathbf{W}_1)=\text{trace}(\mathbf{W}_2)=1\\ \mathbf{W}_1\succeq 0,\ \ \mathbf{W}_2\succeq 0\\ \mathbf{W}_1+\mathbf{W}_2\ \ \text{is rank two}\\ \text{trace}((\mathbf{W}_1-\mathbf{W}_2)A(\mathbf{x}))\geq 0$$
Then, I have trouble achieving the final goal: an SDP with a rank-$1$ constraint and stuck with some bilinear terms. Can anyone provide some next-step suggestions or further hints?