# Differences between objective function definitions in optimization problems

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 $$Z(\omega),\hat{Z}(\rho,\omega) \in \mathbb{C}$$ $$$$\underset{\rho}{\text{arg min}}\quad \frac{\left\lVert Z(\omega) - \hat{Z}(\rho,\omega)\right\rVert^{2}}{\left\lVert Z(\omega) + \hat{Z}(\rho,\omega) \right\rVert^{2}},$$$$ while the second is normalized by only one of them $$$$\underset{\rho}{\text{arg min}}\quad \frac{\left\lVert Z(\omega) - \hat{Z}(\rho,\omega) \right\rVert^{2}}{\left\lVert Z(\omega) \right\rVert^{2}},$$$$ where $$\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?