A Toolkit for Optimizing Functions in Economics

William L. Goffe - University of Southern Mississippi

Optimization has long played a key role in both economic theory and econometrics. While in years past optimization's use was restricted to analytical problems, today economists are increasingly building more sophisticated and complicated models that can only be solved numerically. Unfortunately, "conventional" optimization algorithms used to solve these problems have changed very little in recent years. With restrictive assumptions about the function, these algorithms were designed for the expensive computing power of past years. However, because of their restrictive assumptions, they often fail on today's difficult optimization problems. With today's entry level PCs having the power of mainframes of years past, new approaches to optimization are in order.

One new optimization algorithm, simulated annealing, designed for more robust optimization of difficult optimization problems, was introduced to economists in "Global Optimization of Statistical Functions with Simulated Annealing" W. Goffe, et al., Journal of Econometrics, 1994. Simulated annealing solves difficult optimization problems, at the expense of more computer time.

This work extends and Goffe et al. with "pre-tests" for difficult optimization problems. These will help pick the appropriate optimization algorithm: "heavy duty" algorithms like simulated annealing or genetic algorithms, or quicker, but less robust, conventional algorithms. Currently, there are few methods to aid the researcher in choosing an algorithm appropriate for the problem. These pre-test methods include different ways of sampling and visualizing the function. One, multi-dimensional cross-sections, was used, but not reported, in Goffe et al. Another, sampling the function with "quasi-random" sequences, has been rarely used in economics. Besides using the results of this sampling as initial values, their distribution helps describe the function. Another tests whether the function indeed fulfills the restrictive requirements of "conventional" optimization algorithms (specifically, is the function quadratic in shape). Each method works in different ways, and will provide synergistic information to the researcher. Many of these tests will exploit today's cheap computing power.

Hopefully, with this toolkit, economists and econmetricians will be more able to more easily solve today's complicated problems. In other words, if found useful by "practitioners", this will lead to more time spent doing research, and less time spent in frustration trying to optimize difficult problems.

Scheduled for Session 3.6 Numerical Methods