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Can someone explain the differences between the relative error definitions used as an objective function in a constrained optimization problem? I am trying to understand why I get different values of $\rho \in \mathbb{R}^{3}$ using fmincon in MATLAB.

The first is normalized by the sum of the two complex valued functions $Z(\omega),\hat{Z}(\rho,\omega) \in \mathbb{C}$ \begin{equation} \underset{\rho}{\text{arg min}}\quad \frac{\left\lVert Z(\omega) - \hat{Z}(\rho,\omega)\right\rVert^{2}}{\left\lVert Z(\omega) + \hat{Z}(\rho,\omega) \right\rVert^{2}}, \end{equation} while the second is normalized by only one of them \begin{equation} \underset{\rho}{\text{arg min}}\quad \frac{\left\lVert Z(\omega) - \hat{Z}(\rho,\omega) \right\rVert^{2}}{\left\lVert Z(\omega) \right\rVert^{2}}, \end{equation} where $\left\lVert \cdot \right\rVert^{2}$ is the $L^{2}$-norm over frequency $\omega$.

Does this have something to do with the complex functions $Z(\omega),\hat{Z}(\rho,\omega) \in \mathbb{C}$ inside the norm?

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