Why the objective function in optimization does not follow dimensional rule? I have seen few studies where the objective function tries to minimize two different variables (let's say $a$ and $b$) that are of different dimensions using a superposition rule. For example, this is what they do:
$$f = \min \left[ (a_{estimated} - a_{observed})^2 + (b_{estimated} - b_{observed})^2 \right]$$
What's your comment regarding this practice? Am I missing something in above interpretation that suggests this is mathematically inconsistent?
I thought this could have been mathematically consistent if we divided their difference by their variances, but that would actually go against the minimization principle because a higher variance would artificially decrease the objective function $f$, when it should be opposite. 
 A: It's a hack.
Multi-objective optimization is significantly more complex than optimizing a single objective function (for instance there are whole families of solutions along the Pareto frontier) so it is common practice to combine all of the terms you desire to be small into a single objective, in a more-or-less ad-hoc way.
Usually you use a parameter to scale the two terms:
$$\min_x\ \|f(x)\|^2 + \lambda \|g(x)\|^2$$
where $\lambda$ is chosen to rate the relative "importance" of the two terms. This $\lambda$ can be assigned whatever units are necessary to make the two terms commensurate (which of course does not change the fact that the value selected is often ad-hoc.) Your objective is the special case $\lambda=1$.
EDIT: If you're confused about vector space dimension (rather than unit dimension) note that the optimization problem in the OP is really
$$f = \min \|a_{estimated} - a_{observed}\|^2 + \|b_{estimated} - b_{observed}\|^2$$
for some norm $\|\|$ (usually the two-norm but, increasingly, the one-norm thanks to its sparsification properties.) Some authors write $(v-w)^2$ for vectors $v$ and $w$ to mean $\|v-w\|^2$, which surely causes confusion.
