Chapter 3

GEOMETRIC PROGRAMMING

 Introduction top

In 1961 Clarence M. Zener, Director of Science at Westinghouse, published the first of several papers (1) on a new optimization technique he had discovered while working on the optimal design of transformers. These papers attracted the attention of Professor D.J. Wilde of Stanford University, and in 1963 he described them to the author of this textbook who obtained copies from Dr. Zener. In this work, Dr. Zener used the result called Canchy's arithmetic-geometric inequality which showed that the arithmetic mean of a group of terms always was greater than or equal to the geometric mean of the group, and he was able to convert the optimization of the nonlinear economic model for transformer design to one of solving a set of linear algebraic equation to obtain the optimum. The use of Cauchy's arithmetic-geometric inequality led to the name of geometric programming for the technique.

Two relatively parallel and somewhat independent efforts began to expand and extend the ideas about geometric programming. These were by Zener and colleagues and by Wilde and his students. A professor of mathematics at Carnegie Mellon University, Richard Duffin, began collaborating with Zener to extend the procedure. They were joined by Elmor Peterson, a Ph.D. student of Duffin. In 1967 Duffin, Peterson, and Zener (2) published a text on their work entitled Geometric Programming. In this work the economic model was limited to minimizing the cost of the sum of positive terms.

Wilde and his student Ury Passey (3) developed the theory for negative coefficients and inequality constraints using Lagrange methods. This research went directly into the text Foundations of Optimization, which was published in 1967, and the result is that geometric programming is now applicable to a polynomial economic model with polynomial constraints as equalities and inequalities.

Four other books have followed the publication of those mentioned above. Zener (4) followed with a book entitled Engineering Design by Geometric Programming in 1971. Nijhamp wrote a book on the subject, entitled Planning of Industrial Complexes by Means of Geometric Programming in 1972. The third and fourth volumes, completely covering the theory and application of the subject, were Applied Geometric Programming by Beightler and Phillips(6), published in 1976, and Globally Optimal Design by Wilde (14) published in 1978.

The more important results for this optimization procedure will be described in this book, which will take us through unconstrained polynomial optimization. This will show the advantages and disadvantages of the techniques and how the method capitalizes on the mathematical structure of the optimization problem. Also, this will give those who are interested the ability to proceed with additional material on the topic given in the previously mentioned books without significant difficulty. Beginnning with posynomial optimization, we will then proceed to polynomial optimization. We will find that the global minimum is obtained with posynomials, but only local stationary points are obtained when the economic model is a polynomial. Our approach will follow that of Wilde and Passey (3). On seeing Zener's work, they were able to obtain the same results from the classical theory of maxima and minima and extend this to polynomials. Consequently, we will use the classical theory to develop geometric programming, although this will not describe Zener's original development using the geometric-arithmetic inequality. However, it will require less effort to obtain the final result, since the background arguments associated with the geometric-arithmetic inequality will not be required, and the results for polynomial optimization will follow directly from posynomial optimization.