Function optimization was applied in Chapter 4, Simulation of Random Numbers to find the maximum of a normal density divided by a Cauchy density as well as to find the extreme of a Beta distribution function. In this chapter, we will concentrate on two-dimensional problems and note that the mentioned methods can be extended to multi-dimensional problems. To convey a feeling of how optimization methods work, we start with a story set in the Austrian Alps.
When I wrote these lines we suddenly had foggy weather in Austria. And I imagined the scenario of a guy from Australia visiting Austria. Note that kangaroos exist only in the zoo in Austria, and that 70 percent of Austria is covered by mountains (part of the Alps). Assume that the Australian guy has no prior information (no maps, no conversations, no guide at all) and he starts to climb the mountains. These mountains represent, in other words, a three-dimensional complex non-concave...