A comment on EDSTAT-L reminded me about this important issue. Many problems in Statistics involve finding an optimal point of a rather complex and messy function. In some situations, the optimal point is the largest value for this function and in other situation, the optimal point is the smallest value. The simplest and best known example of this is regression. This approach tries to estimate a line (or sometimes a more complex curve) that is "close" to most of the data. The approach, least squares, tries to minimize the sum of squared deviations between the line and the data. Even simple problems such as estimating a median can be recast in terms of minimizing distance.
Optimization using a computer is a rather difficult and complex process because an approach that works well for one set of problems may perform poorly for another set. A nice tutorial on the state of the art for optimization is:
- A Tutorial on MM Algorithms. Hunter DR, Lange K. The American Statistician 2004: 54(1); 30-37.
You can find an earlier version of this page on my original website.