Consider the following optimization: \begin{equation} \max_{\boldsymbol{x}} \frac{\boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}}{\|\boldsymbol{x}\|^2} + \frac{\boldsymbol{x}^T \boldsymbol{B} \boldsymbol{x}}{\boldsymbol{x}^T \boldsymbol{A} \boldsymbol{x}} , \end{equation} where $\boldsymbol{x}$ is $n\times1$ real vector, $\boldsymbol{A}$ and $\boldsymbol{B}$ are $n \times n$ positive definite matrices.

The first term is a standard Rayleigh quotient and the second is a generalized Rayleigh quotient, each of which individually has known closed form solution (in terms of eigenvalues). However, I do not know if the solution of the above sum also admits a closed form solution or not.

Any advice would be greatly appreciated.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.