# 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.