1D optimization in R
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Introduction
R provides functions for both onedimensional and multidimensional optimization. The second topic is much more complicated than the former (see e.g. Nocedal 1999) and will be left for another day.
A convenient function for 1D optimization is optimize()
, also known as optimise()
. Its first argument is a function whose minimum (or maximum) is sought, and the second is a twoelement vector giving the range of values of the independent variable to be searched. (See ?optimize
for more.)
Application
As an example, consider the phase speed of deep gravitycapillary waves, which is given by $\omega/k$ where $\omega$ is the frequency and $k$ is the wavenumber, and the two are bound together with the dispersion relationship $\omega^2=gk+\sigma k^3/\rho$, where $g$ is the acceleration due to gravity, $\sigma$ is the surface tension parameter (0.074 N/m for an airwater interface) and $\rho$ is the water density (1000 kg/m^3 for fresh water). This yields wave speed given by the following R function.
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phaseSpeed < function(k) { g < 9.8 sigma < 0.074 # waterair rho < 1000 # fresh water omega2 < g * k + sigma * k^3/rho sqrt(omega2)/k } 
It makes sense to plot a function to be optimized, if only to check that it has been coded correctly, so that is the next step. Readers who are familiar with gravitycapillary waves may know that the speed is minimum at wavelengths of about 2 cm, or wavenumbers of approximately $2\pi/0.02=300$; this suggests an x range for the plot.
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k < seq(100, 1000, length.out = 100) par(mar = c(3, 3, 1, 1), mgp = c(2, 0.7, 0)) plot(k, phaseSpeed(k), type = "l", xlab = "Wavenumber", ylab = "Phase speed") 
The results suggest that the range of $k$ illustrate contains the minimum, so we provide that to optimize()
.
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o < optimize(phaseSpeed, range(k)) phaseSpeed(o$minimum) ## [1] 0.2321 
This speed is not especially fast; it would take about a heartbeat to move past your hand.
Exercises

Use
str(o)
to learn about the contents of the optimized solution. 
Use
abline()
to indicate the wavenumber at the speed minimum. 
Try other functions that are of interest to you, e.g. find the angle that maximizes $\sin\theta\cos\theta$, which yields the throwing angle that achieves furthest distance in frictionless air over flat terrain.

Use the multidimensional optimizer named
optim()
on this problem.
References
Jorge Nocedal and Stephen J. Wright, 1999. Numerical optimization. Springer
series in operations research. SpringerVerlag, New York, NY, USA.
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