Answer by Rodrigo de Azevedo for Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$

The smallest eigenvalue can be found (approximately) via the following semidefinite program (SDP).

$$ \begin{array}{ll} \underset {t} {\text{maximize}} & t \\ \text{subject to} & \operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top \succeq t \, {\bf I}_n \end{array} $$