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I want to solve the following maximization problem in $\mathbf{X}\in {\mathbb{R}}^{{m} \times {n}}$

\begin{eqnarray} \begin{split} \quad\max_\mathbf{X} \frac{\mathbf{Tr}(\mathbf{X}^\top \mathbf{Q} \mathbf{X})}{\mathbf{Tr}((\mathbf{X-A})^\top \mathbf{P} (\mathbf{X-A}))} \end{split} \end{eqnarray}

where $\mathbf{A}\in {\mathbb{R}}^{{m} \times {n}}$, $\mathbf{P}\in {\mathbb{R}}^{{m} \times {m}}$ and $\mathbf{Q}\in {\mathbb{R}}^{{m} \times {m}}$ and $\mathbf{P}$, $\mathbf{Q}$ are positive definite matrix.

It is well known that if $\mathbf{A}=\mathbf{O}$, it is easily solved by eigenvalue problem. However, I am not sure how to solve such a problem.

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  • $\begingroup$ I am not sure if you require definiteness of the matrices $P$ and $Q$ for the maximum to exist. $\endgroup$ – nicoguaro♦ Jul 23 '19 at 22:03
  • $\begingroup$ Wouldn't in the limit of $X$ to $A$ yield positive infinity for the cost function if $\text{Tr}(A^\top Q\,A)>0$, thus a maximum? Also I believe that it should be that $X,A\in\mathbb{R}^{n\times m}$ (so switching $m$ and $n$)? $\endgroup$ – fibonatic Jul 24 '19 at 3:50

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