Based on my on-again-off-again internet search of optimization problems over the past few years, the two terms seem to cover largely the same way of representing problems, if not the identically. I'm not talking about solver engines in the back end, but rather, the mathematical representation.

P.S. Both terms seem to also overlap with AMPL's model-based optimization, but I'm cautious about using that term because all decision aid methods are based on a model.


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"Algebraic modeling" can presumably mean algebraic models of any sort of system for any purpose. I see it used primarily if not exclusively for optimization problems, but that's largely because I tend to look at only optimization problems. I would think that, for example, formulas for expected waiting times in queuing systems could be considered algebraic modeling.

"Mathematical programming" refers to algebraic models specifically for optimization. So I would consider it a narrower, more precise phrase than "algebraic modeling".

AMPL is a platform built around an algebraic modeling language for mathematical programming.

  • $\begingroup$ Thank you, prubin. Based on your distinction of the two terms, I did another internet search of the two terms. By and large, I can see the distinction that you drew. On the note about AMPL, I've played around with it, but my "P.S." was about its use of the term "model-based optimization". The reason why I'm cautious about using that term is because all optimizations are based on models, not just the ones represented in AMPL and not just mathematical programming models. $\endgroup$ Feb 24 at 6:33
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    $\begingroup$ There's some truth to your observation that all optimizations are based on models, but not all are based on algebraic models. For instance, one can argue that gradient descent is not, nor are evolutionary algorithms. I'll stop picking semantic nits there because I don't want anyone thinking I've turned into a lawyer. :-) $\endgroup$
    – prubin
    Feb 24 at 18:49
  • $\begingroup$ Those are some of the ones I had in mind. I don't think of it as lawyering. It's more about accurate portrayal. A more accurate term would be algebraic modelling optimization because it doesn't encompass all the other forms of optimization. However, that would just boil down to mathematical programming, which has drawbacks in terms of marketing. $\endgroup$ Feb 25 at 23:42
  • $\begingroup$ "Mathematical programming" does have the danger of being interpreted as "writing code that does some sort of math" (evaluating definite integrals, say). $\endgroup$
    – prubin
    Feb 26 at 3:51
  • $\begingroup$ I agree that it is a possible misinterpretation of those totally new to the area, but at least Google shows a pretty comprehensive consensus on its formal definition. Operational researchers won't misinterpret that. Data analysts and programmers might wonder about such an odd term, but can quickly figure out its precise meaning. I worry that coining a very inaccurate term like model-based optimization causes way more confusion because it is contrasted with "method-based" optimization (everything that isn't mathematical programming). $\endgroup$ Feb 26 at 4:26

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