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I want to optimize a sum of complex frequency responses (complex sum is SPL, sound pressure level as a complex function of distance) of a number of loudspeakers, say 10. There is a reference, desired SPL curve given (objective function), which represents the magnitude of the sum of 10 frequency responses. I have read a number of papers how to do optimization using Lagrange Multipliers, however, did not find many papers (only one) about optimization when the functions are complex (Re and Im parts).

Any help or pointing to a good references, examples would be appreciated.

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I found the following in Charles Therrien's "Discrete Random Signals and Statistical Signal Processing" in one of the Appendicies.

Say you have the function $Q(a)$ you wish to minimize such that $C(a)=0$, where $C(a)$ may be complex valued and $a$ may be a complex vector. The constraint really represents two real-valued constraints. $$C_r(a)=0,\qquad C_i(a)=0, $$ where $C_r(a)$ and $C_i(a)$ represent the real and imaginary components. The Lagrangian can be written as: $$\mathcal{L}=Q(a) +\mu_1C_r(a)+\mu_2C_i(a), $$ where $\mu_1$ and $\mu_2$ are real valued.

This can also be written more compactly using a single complex parameter $\lambda=\lambda_r+j\lambda_i$ as $$\mathcal{L}=Q(a) +\lambda C(a)+\lambda ^*C^*(a) $$ where $\lambda^*$ and $C^*(a)$ denote the complex conjugate of the quantities. The reason for this is that $$ \lambda C(a)+\lambda ^*C^*(a)=2Re[\lambda C(a)]=2\lambda_rC_r(a)-2\lambda_iC_i(a) $$ and letting $\mu_1=2\lambda_r$ and $\mu_2=-2\lambda_i$, then the two Lagrangians are seen to be equivalent.

Note - to apply the gradient to a real valued function with respect to a complex parameter you should also be familiar with Brandwood "A complex gradient operator and its application in adaptive array theory" IEE Proceedings 130(1) 11-16, Feb, 1983. This essentially treats the variables $a$ and $a^*$ as two separate variables and you only have to take the derivative wrt to one of them to perform a minimization.

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Any optimization of
$\qquad$ min $f(z), \ z$ a vector of $n$ complex variables
is equivalent to
$\qquad$ min $f( x, y ), \ z = x + i \, y: 2n$ real variables.
The complex form is often more natural than the $2n$ real — saves thought and ink. But here, the real Lagrangian
$\qquad f( x, y ) + \lambda_{re} \, \text{constraint}_{re} + \lambda_{im} \, \text{constraint}_{im} $
is simpler (for me).

See also lagrangian-multipliers-in-complex-optimization on math.stack.

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