Maximizing $frac{X_{11}det(T)}{T_2^2+T_4^2}$ subject to ${TXT^T}_{ii}leq P_i$












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begin{array}{ll} text{maximize} & frac{X_{11}det(T)}{T_2^2+T_4^2}\ text{subject to} & {TXT^T}_{ii}leq P_iend{array}
where $Tinmathbb{R^{2times2}}$, $T=begin{bmatrix}T_1 & T_2\T_3 &T_4end{bmatrix}$, $T$ is invertible, $Xinmathbb{R^{2times2}}$, $X=begin{bmatrix}X_{11} & X_{12}\X_{12} &X_{22}end{bmatrix}>0$ (positive definite).



We need to maximize in terms of $P_1$ and $P_2$, which are given. I have no idea how to tackle this question, even when $T$ is unitary, we just need to maximize $X_{11}$, it turns out not straightforward.










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    $begingroup$


    begin{array}{ll} text{maximize} & frac{X_{11}det(T)}{T_2^2+T_4^2}\ text{subject to} & {TXT^T}_{ii}leq P_iend{array}
    where $Tinmathbb{R^{2times2}}$, $T=begin{bmatrix}T_1 & T_2\T_3 &T_4end{bmatrix}$, $T$ is invertible, $Xinmathbb{R^{2times2}}$, $X=begin{bmatrix}X_{11} & X_{12}\X_{12} &X_{22}end{bmatrix}>0$ (positive definite).



    We need to maximize in terms of $P_1$ and $P_2$, which are given. I have no idea how to tackle this question, even when $T$ is unitary, we just need to maximize $X_{11}$, it turns out not straightforward.










    share|cite|improve this question











    $endgroup$















      -1












      -1








      -1





      $begingroup$


      begin{array}{ll} text{maximize} & frac{X_{11}det(T)}{T_2^2+T_4^2}\ text{subject to} & {TXT^T}_{ii}leq P_iend{array}
      where $Tinmathbb{R^{2times2}}$, $T=begin{bmatrix}T_1 & T_2\T_3 &T_4end{bmatrix}$, $T$ is invertible, $Xinmathbb{R^{2times2}}$, $X=begin{bmatrix}X_{11} & X_{12}\X_{12} &X_{22}end{bmatrix}>0$ (positive definite).



      We need to maximize in terms of $P_1$ and $P_2$, which are given. I have no idea how to tackle this question, even when $T$ is unitary, we just need to maximize $X_{11}$, it turns out not straightforward.










      share|cite|improve this question











      $endgroup$




      begin{array}{ll} text{maximize} & frac{X_{11}det(T)}{T_2^2+T_4^2}\ text{subject to} & {TXT^T}_{ii}leq P_iend{array}
      where $Tinmathbb{R^{2times2}}$, $T=begin{bmatrix}T_1 & T_2\T_3 &T_4end{bmatrix}$, $T$ is invertible, $Xinmathbb{R^{2times2}}$, $X=begin{bmatrix}X_{11} & X_{12}\X_{12} &X_{22}end{bmatrix}>0$ (positive definite).



      We need to maximize in terms of $P_1$ and $P_2$, which are given. I have no idea how to tackle this question, even when $T$ is unitary, we just need to maximize $X_{11}$, it turns out not straightforward.







      matrices optimization






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      share|cite|improve this question













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      edited Jan 3 at 7:36









      Saad

      19.7k92352




      19.7k92352










      asked Jan 3 at 7:16









      LeeLee

      328111




      328111






















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