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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.

A.Γ.
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Cort Ammon
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  • I guess one way to arrive at a definition would be to think of a specific situation in which you want to optimize several values at once. – Javier Feb 04 '18 at 16:46
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    You probably learned this already, but in multi-criteria optimization problems, people often search for Pareto optimal solutions. The approach you mentioned using $S(e,m)$ finds Pareto optimal solutions, and by varying $k_1$ and $k_2$ we can explore the Pareto frontier. – littleO Feb 04 '18 at 16:48
  • @Javier I cannot think of one, myself. I've been thinking about it for a while. However, that doesn't mean that someone didn't think of a meaning and put enough formalism behind it to be of interest. – Cort Ammon Feb 04 '18 at 16:48
  • Multi-criteria optimization problems arise commonly in practice. For example, in radiation therapy, we want to minimize the amount of radiation delivered to various organs subject to a constraint that the tumor receives a certain minimum amount of radiation. So we have one objective function for each organ that we are trying to protect. Radiation treatment planning systems often use multi-criteria optimization methods that explore the Pareto frontier then present the clinician with a selection of Pareto optimal treatment plans. – littleO Feb 04 '18 at 16:51
  • @littleo pareto optimality it is the direction I am looking. It still calls for a pairwise mapping onto a goodness measure. It is possible that that is indeed the closest mathematics gets to what I am thinking, but I'm curious if there is something closer to my Ill defined question – Cort Ammon Feb 04 '18 at 16:58
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    For optimization to make sense you need some kind of ordering on possible outputs. There isn't a consistent ordering on $\Bbb{R}^n$ for $n>1$, so it would be hard to define 'best' in that space. To get a subset of the objective space that does have a consistent ordering, I suspect you'd have to do something equivalent to mapping into $\Bbb{R}$ anyway. – Sort of Damocles Feb 04 '18 at 17:02

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