3
$\begingroup$

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.

enter image description here

$\endgroup$
0

1 Answer 1

6
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ thanks, that's the trick I was missing! $\endgroup$ Aug 22, 2022 at 8:08
  • $\begingroup$ I deleted my previous comments now that we have this nice answer. $\endgroup$ Aug 22, 2022 at 11:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.