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 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 Four other books have followed the publication
of those mentioned above. Zener (4) followed with a book entitled 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. |