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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.

Golabi

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