You are absolutely correct that some sort of lexicographic ordering could exist: https://en.wikipedia.org/wiki/Lexicographical_order#Finite_s...

If such an ordering did exist, then we could certainly apply that ordering to sort results from the vector objective function so as to find the "answer" to the multicriteria problem. The Wikipedia article on multiobjective optimization discusses this strategy: https://en.wikipedia.org/wiki/Multi-objective_optimization#A.... On that note, lemme throw a shout out to the wonderful person who took the time to write that Wikipedia article - it is outstanding.

Given that, such an ordering may not be appropriate in all circumstances. Sorting objective vectors from the function suggested in this post would first sort by "time to destination" and then break ties in "time to destination" with "cost of trip". That would mean that (1, 1000) < (1.0000001, 2), but I think most people would be willing to arrive 0.0000001 hours later to save 998 dollars. The flexibility in interpreting the vector objective and making tradeoffs is why the standard lexicographic ordering is not always appropriate.

Does that help?

> most people would be willing to arrive 0.0000001 hours later to save 998 dollars

Unfortunately, if you don't know how your model behaves, you can't tell when you're setting γ whether you're actually going to get a solution that's way past the point of diminishing returns. You may not even be able to tell after the fact. This is one of the reasons for doing sensitivity analysis on γ. (Your ε-constraint scalarization helps with this problem.)

You make a perfectly accurate point that, in practice, it is unlikely one would be able to make such a prediction without significant info about the model. The above comment was meant in more of a post hoc "we've executed our multicriteria optimization, approximated our Pareto frontier, now let's make a decision" sense, not within a specific scalarization context. Indeed sensitivity analysis on the gamma parameter is very important; I tried to hint at that by showing that the interplay of the choice of currency and the gamma on the optimal value is nontrivial (though predictable since this is just a toy problem).

Do you happen to have any references talking about such sensitivity analysis on scalarization parameters? I would love to add them to my reading list. Thanks.

I don't have any references -- I'm going off something one of my committee members said. But the basic idea is just that, if your weight is 0.2, you want to make sure you also obtain solutions for 0.18 and 0.22, to give you some sense of whether you're on the edge of a cliff. This is pretty cheap in general for convex optimization problems because you've already got a solution that should be close to optimal for the new weight.

Good call - if the problem is well behaved then small changes in gamma should be able to use the previous solution as an initial guess. And I absolutely agree with the robustness idea you're talking about; I was hinting at it when I was talking about the impact of choice of currency. For a well scaled problem there is a consistent and well-behaved impact on the solution for small changes in gamma. But when the currency was changed to RMB, small changes in gamma no longer had a consistent impact on the optimum.

If you change the basis of R_n (e.g., you rotate the axes, or you can do), you get a different result by this comparison. If there's no intrinsic reason for using a particular basis, you get different results with this ordering. In other words, you're more or less being completely arbitrary.

this implies you're giving weight to the first element then the second.

let's say you were optimizing something to be pretty A, delicious B, soft C.

you tune the system and evaluate the prettiness, deliciousness and softness.

you tuned it several times and got three products (A, B, C) of (1,2,3), (2,1,3), (3,2,1) - concrete values are correct evaluations. how exactly do you choose the best one, is prettiness more important?

pareto efficiency come to mind - also discussed in the article. [1]

[1] : https://en.wikipedia.org/wiki/Pareto_efficiency

Consider a simpler case of two dimensional real vectors. Lets consider the vector space of R^2, since all finite dimensional vector spaces of same dimensionality are isomorphic, R^2 is isomorphic to the complex numbers.

How would you order the complex numbers?

> How would you order the complex numbers?

You can use any order that can be defined on R^2. It's just that you can show that this order cannot satisfy the axioms of an ordered field.

Good call. That's a simpler example than where I was going.