# How can I formulate a minimization problem with objective values differing in scale?

I am trying to solve a standard optimization problem,

minimize distance1 + distance2

Where distance1 and distance2 are different distance measures for comparing time-series data. Now, both distance measures are on different scales.

Let's say I know the ranges of each distance, measures, e.g., distance1 ranges from [0, 1000] and distance2 ranges from [0,1]. In this case, I could use tools such as min max normalization.

Now, what if I don't know the ranges a priori? How can I combine both distance measures into a single distance measure without comparing apples to oranges?

• A multiplicative factor, say on distance2, should account for both differences in scale and importance. Jul 21, 2022 at 17:35
• The choice of scale is going to be a heuristic. Even if $d_1 \in [0,1000]$, what if 99% of values are below 50, whereas $d_2$ is uniformly distributed? Another heuristic is choosing $d_1+d_2$ as opposed to say $\sqrt{d_1^2+d_2^2}$. I don't think there's any problem independent justification for these choices. If the "optimal" solution is supposed to be "best" is some way in the real world, and you have some independent way of measuring that, you could investigate how the scale, and choice of norm, impact the quality of the optimal solution.
– Joe
Jul 22, 2022 at 9:23