Rewriting quadratically-constrained optimization problem as a semidefinite program

Question

Suppose $A,H$ are positive definite matrices and $\alpha,t$ are scalars. Is there a way to massage the following problem into a form suitable for a specialized solver?

$$\begin{array}{ll} \underset{\alpha,t}{\text{minimize}} & t\\ \text{subject to} & A-\alpha AH - \alpha HA + \alpha^2 HAH \prec t I\end{array}$$

It "almost" works as SDP, except for the $\alpha^2$ term in the constraint. It comes down to finding $\alpha$ such that corresponding quadratic form fits in a small circle, needed for guaranteeing stability of iteration.

Answer 1

With a factorization such as $HAH = R^TR$ you can apply a Schur complement and use $\begin{pmatrix}tI+\alpha (AH+HA)-A & \alpha R^T\\\alpha R & I\end{pmatrix} \succeq 0$.