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I have multiple objectives, such as $f(\mathbf{x})$, $g(\mathbf{x})$, and $h(\mathbf{x})$.

I would like to find a set of $x$ that can $\underset{\mathbf{x}}{argmin} [ f(\mathbf{x}) + g(\mathbf{x}) + h(\mathbf{x}) ]$.

However, the three objectives have quite different scales. For instance the output of $f(\mathbf{x})$ is normally in [1000, 5000], while that of $g(\mathbf{x})$ is in [0, 1].

Does this negatively impact the oprimization? To compute the $argmin$, should I normalize them to an identical scale?

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  • $\begingroup$ It depends on what the various objectives mean. When there's no meaningful interpretation, it's common to add tunable parameters as coefficients: $f(\mathbf{x}) + \lambda g(\mathbf{x}) + \gamma h(\mathbf{x})$. By changing $\lambda$ and $\gamma$, you change the relative importance of $f$, $g$, and $h$ in the overall optimization. $\endgroup$ – tddevlin Apr 11 at 17:58
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As mentioned in a comment by @tddevlin, it depends on what those objectives are. I give you an example. Suppose I'm solving a vehicle routing problem and I want to minimize both cost and number of vehicles. Then, my $f(x)$ is cost and $g(x)$ is the number of trucks, which both fit the profile that you mentioned. Now, if I prefer costs over the number of trucks, I don't even need to add any weight to my $f$ and $g$ functions. In fact, I can use them exactly as is and know that because of the small scale of $g$ compared to $f$, always $f$ will be preferred first, which is the desired behavior. That's why it matters what are your objectives.

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