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Where is the line when an approach is called multi-objective optimization? For example:

Problem

Presume I want to optimize an optimization problem, for example nurse rostering, with 2 soft constraints:

  • Constraint A) for employee happiness: number of rejected "day off requests"
  • Constraint B) for service quality: number of shifts that match the employee's affinity

Now, I don't know yet if A) is a higher priority than B) or visa versa.

Solution

If - instead of running an optimization to come up with a pareto front - I run a normal solver twice:

  • Solve once with weights A=1 and B=1000
  • Solve again with weights A=1000 and B=1

and then allow the user to pick from these two simulations the solution which he/she prefers.

Question

Would you still call this approach multi-objective solving? What does the literature say?

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  • $\begingroup$ Are you considering "simulation" as synonymous with "single-objective optimization"? What is the fundamental difference between your "simulations" and a Pareto front that would consist of those two points plus additional similar ones? $\endgroup$
    – nanoman
    Oct 15 at 5:27
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There's a fair sized body of research in interactive multiobjective optimization, and while I'm not familiar with most of it, I think this would fit right in. Decades ago, I (vaguely) remember two of my colleagues looking at an interactive approach for multiobjective LPs, in which they would combine the criteria using a weighted sum, solve, show the individual objective values to the user, let the user adjust the weights by increasing/decreasing the importance of various criteria, solve again and iterate ad nauseam (meaning they would continue until the decision maker fatigued and begged them to stop). That was considered multiobjective optimization, so I assume your approach would be.

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    $\begingroup$ Interesting. So we've been doing multi-objective optimization for years - by playing with constraint weights and running simulations basically, but never called it that because it doesn't use a pareto front... This also answers my other question! I should have asked if anyone has ever seen pareto front optimization survive in production. :) Thanks Paul! $\endgroup$ Oct 14 at 19:40
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tl;dr The term you're looking for is sensitivity analysis.


Would you still call this approach multi-objective solving? What does the literature say?

Trying different possible parameters to form a portfolio of options is called doing a sensitivity analysis.

For example, say you're an engineer designing a light-bulb. You know you want it to last a long time, be power-efficient, cheap, and all that good stuff – though you don't want to make assumptions about the trade-offs between those things, e.g. if making it 1% cheaper at the cost of being 1% less efficient is a good trade or not. Then you can do a sensitivity-analysis: generate a portfolio of results corresponding to various plausible trade-offs.

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  • $\begingroup$ Good point! The approach is indeed are exactly the same as a sensitivity analysis. $\endgroup$ Oct 15 at 9:02
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Let me add to the existing answers the following result [1]:

Let $f_1,f_2$ be two convex differentiable functions on $\mathbb{R}^N$, and consider the corresponding multiobjective optimization problem. Then the set of weak Pareto solutions is exactly equal to $$ \bigcup\limits_{\alpha,\beta \geq 0} {\rm{argmin}}~\alpha f_1 + \beta f_2.$$

In other words, any minimizer of a weighted sum of your functions is a weak Pareto equilibrium, and reciprocally. If you get rid of the convexity assumption on the functions, you lose the reciproque.

In your context, it means that with your chosen weights A and B, you are choosing two specific Pareto equilibriums among all the possible ones. I would consider this to be multiobjective optimization.

[1] : Theory of vector optimization, Dinh The Luc, 1988. See Proposition IV.2.10.

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