Let $A$ and $B$ be two given hermitian positive semi-definite matrices, then what is the solution for \begin{align} \max_{x\neq 0}\frac{x^HAx}{x^HBx+1}. \end{align} I am looking for closed form solutions. If the denominator didn't have that $1$, this is standard generalized rayleigh quotient and would be unbounded.

I know how to solve it numerically. The trick is to re-write it as \begin{align} \max_{x,t}~&t\\ \text{s.t.}~~&x^H(A-tB)x>=t \end{align} Then find the largest $t$ such that there exists a $x$ which satisfies $x^H(A-tB)x>0$. A Bi-section search on $t$ will do the job.

I have already asked this question before in math stack exchange. Since I didn't get any answers, I thought I will post it up here.

Let $A$ and $B$ be two given hermitian positive semi-definite matrices, then what is the solution for

$$\max_{x\neq 0}\frac{x^HAx}{x^HBx+1}.$$

I am looking for closed form solutions. If the denominator didn't have that $1$, this is standard generalized rayleigh quotient and would be unbounded.

I know how to solve it numerically. The trick is to re-write it as

$$\max_{x,t}~t\\ \text{s.t.}~~x^H(A-tB)x \ge t$$

Then find the largest $t$ such that there exists a $x$ which satisfies $x^H(A-tB)x\gt0$. A Bi-section search on $t$ will do the job.

I have already asked this question before in math stack exchange. Since I didn't get any answers, I thought I will post it up here.