I'm exploring a philosophical question which lead me towards the idea of optimizing in multiple dimensions, not just for inputs, but for the evaluated result as well. A physical example might be for maximizing "energy and momentum." All of the multidimensional-optimization approaches I am aware of start by doing some mapping onto one dimension so that we can use the traditional one dimensional definitions of maximum (or minimum) to do the optimization. In the "energy and momentum" example, that might be done with "maximize $S(e, m) = k_1e + k_2m$, where $e$ is the energy and $m$" (and the constants have the units to make that make sense). Doing so maps a f: ℝⁿ -> ℝ² problem to a f: ℝⁿ -> ℝ, so that the concept of "maximize" makes sense. We also might try to maximize the gradient of $S$, which has the same effect of mapping that 2 dimensional space into 1.
I can't think of any meaning for "optimize" which work in higher dimensions without such a reduction, which is interesting for the question that I'm posing to myself. I'm wondering if there is a sensical meaning for such a concept.
My question to Mathematics.SE is whether there is any prior art in mathematics which has sought to define "optimize" in a multidimensional system without first reducing the outputs of the function to a single dimension such that the traditional meanings of "maximum" and "minimum" have a meaning.
Note: I phrased this question using ℝ because it is the most familiar to me. If there is an approach which works over a different class, but doesn't work on ℝ for some reason, I would be interested in that as well.