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I am looking for a function of on an interval with several local optima that appears in some mathematical model and which you can at least imagine that you want to optimize.

I am teaching a calculus course right now, and as usual there is a short chapter on optimization. I wanted to have give a list of examples of functions on an interval $[a,b]$ that you might want to optimize and which display different behavior.

Now I find it easy to give examples of monotone functions from applications and it is also not difficult to find good examples of applied problems where the functions you optimize is U-shaped.

But I actually got stuck trying to find good examples of more complex behavior with several local optima with an applied motivation.

Ideally the function I am looking for should come from a real application in science or otherwise and you should at least be able to imagine that you want to optimize it. It has to be a single-variable function and it would be better the less knowledge one needs to understand what the function is modelling.

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    $\begingroup$ Would sinusoidal motion work? The height of an object attached to a wheel, gear, or possibly piston? Too short to be an answer, but hopefully helpful. "Damped oscillation" would also be a fantastic example, e.g. sin(x)*e^-x. $\endgroup$
    – Opal E
    Dec 10, 2015 at 21:01
  • $\begingroup$ Yes, I suppose that could be a good example. The more the merrier however, so keep them coming. $\endgroup$
    – Johan
    Dec 11, 2015 at 17:39

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The graph below shows the distance from the origin (black dot) to the (blue) convex polygonal curve, which alternately has vertices (red) on the unit-radius semicircle (brown), and slightly inside that semicircle. The distance $d$ is plotted with respect to $\theta$, the angle w.r.t. the $x$-axis. You can see it is multimodal.

          MultiModalDists
Slight perturbations of the vertices (red) would break the symmetries, if you prefer more variation in the peaks and valleys.

This multimodal distance function $d(\theta)$ plays a role in algorithms to find the diameter of a convex polygon.

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Maybe consider a sinusoid(ish) curve superimposed on a general trend. A place where you can see this is in business (e.g. volumes of production) where there is some general trend over time but there is SEASONALITY of the business.

Some examples:

  1. ND oil production from 2009-2014 (low during winters because of weather issues but growing over time with development of the field at high prices).

  2. Wal-mart expansion over last couple decades (maybe they have slowed recently, donno) but if you look at monthly data, obviously it is higher in the holidays and lower in January.

[But I struggle with why you would want to find max/min. I guess you can come up with some rationales like hiring for Walmart or takeaway/service support for oil business. Not a perfect example, but maybe there is some insight in there, that is developable.]

P.s. The mechanical examples to use are higher order polynomials. Easy for them to work on and has an added benefit of getting them to think about the analytical geometry issues of how order relates to number of min/max points. But I am struggling to think of a good applied example of a cubic or higher, especially with noticeable wiggles. But still, they get value even from just crunching the math even if it is hard to think of an application.

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