3.6 Case Study: Finding the Minimum of a Function

This section shows how you can write a SAS/IML module to minimize a continuous function of one variable on a closed interval. To make the problem easier, assume that the function is unimodal, meaning that it has a single minimum on the interval.

There are many techniques for finding the minimum of a continuous function on a closed interval. The technique presented in this section is known as the golden section search (Thisted 1988). The basic idea is to bracket the minimum, meaning that you must find three points a < x₁ < b such that the value f (x₁) is less than the function evaluated at either endpoint. In the golden section search, the initial interval [a, x₁] is smaller than the interval [x₁, b]. Therefore, a new point x₂ [x₁, b] is chosen and the function is evaluated at x₂. If f (x₁) < f (x₂), then a < x₁ < x₂ is a new smaller interval that brackets the minimum. Otherwise, x₁ < x₂ < b is a new smaller interval that brackets the minimum. In either case, the process repeats with the new smaller interval. This process is schematically indicated in Figure 3.24.