Restricted optimization using Lagrange multipliers: Difficult problem

My problem is to maximize the following function:
$$ frac { left langle overrightarrow { omega} _ {m} ^ {^ {^ ddagger}} textbf {T} _ {12} overrightarrow { omega} _ {s} right rangle} { sqrt { left langle overrightarrow { omega} _ {m} ^ {^ {^ ddagger}} textbf {T} _ {11} overrightarrow { omega} _ {m} right rangle left langle overrightarrow { omega} _ {s} ^ {^ {^ {ddagger}} textbf {T} _ {22} overrightarrow { omega} _ {s} right rangle}} $$

where

  • $ T_ {11} $, $ T_ {12} $Y $ T_ {22} $ They are 3 by 3 matrices.
  • $ T_ {11} $ Y $ T_ {22} $ They are Hermitian sd-positive matrices.
  • $ T_ {12} $ It is not.
    These matrices contain known data.
    While $ omega_m $ Y $ omega_s $ are two vectors that depend on each of 3 unknown variables, ($ beta_m $ , $ chi_m $, $ delta_m $), Y$ beta_s $ , $ chi_s $, $ delta_s $) respectively as:
    $$ overrightarrow { omega_i} = begin {pmatrix} cos alpha \ without alpha.cos beta_i.e ^ {j delta_i} \ without alpha.sin beta_i.e ^ {j chi_i } end {pmatrix} $$
    NB: $ alpha $ It is a known constant.

And for reasons of speed and efficiency, I want to use restricted Lagrangian optimization. My problem is that the formula contains 6 variables, which means 6 partial derivatives that contain large formulas with cos and no nests. The formula seemed too complex to me. I could not implement it. Any suggestions, help, ideas are welcome, I will be very grateful for the help.