In 1961, C. Zener, then Director of Science at Westinghouse Corpora tion, and a member of the U. S. National Academy of Sciences who has made important contributions to physics and engineering, published a short article in the Proceedings of the National Academy of Sciences entitled" A Mathe matical Aid in Optimizing Engineering Design. " In this article Zener considered the problem of finding an optimal engineering design that can often be expressed as the problem of minimizing a numerical cost function, termed a "generalized polynomial," consisting of a sum of terms, where each term is a product of a positive constant and the design variables, raised to arbitrary powers. He observed that if the number of terms exceeds the number of variables by one, the optimal values of the design variables can be easily found by solving a set of linear equations. Furthermore, certain invariances of the relative contribution of each term to the total cost can be deduced. The mathematical intricacies in Zener's method soon raised the curiosity of R. J. Duffin, the distinguished mathematician from Carnegie Mellon University who joined forces with Zener in laying the rigorous mathematical foundations of optimizing generalized polynomials. Interes tingly, the investigation of optimality conditions and properties of the optimal solutions in such problems were carried out by Duffin and Zener with the aid of inequalities, rather than the more common approach of the Kuhn-Tucker theory.
Author(s): M. Avriel (auth.), Mordecai Avriel (eds.)
Series: Mathematical Concepts and Methods in Science and Engineering 21
Edition: 1
Publisher: Springer US
Year: 1980
Language: English
Pages: 460
Tags: Convex and Discrete Geometry
Front Matter....Pages i-x
Introduction....Pages 1-4
Geometric Programming in Terms of Conjugate Functions....Pages 5-30
Geometric Programming....Pages 31-94
Optimality Conditions in Generalized Geometric Programming....Pages 95-105
Saddle Points and Duality in Generalized Geometric Programming....Pages 107-133
Constrained Duality via Unconstrained Duality in Generalized Geometric Programming....Pages 135-142
Fenchel’s Duality Theorem in Generalized Geometric Programming....Pages 143-149
Generalized Geometric Programming Applied to Problems of Optimal Control: I. Theory....Pages 151-163
Projection and Restriction Methods in Geometric Programming and Related Problems....Pages 165-182
Transcendental Geometric Programs....Pages 183-202
Solution of Generalized Geometric Programs....Pages 203-226
Current State of the Art of Algorithms and Computer Software for Geometric Programming....Pages 227-261
A Comparison of Computational Strategies for Geometric Programs....Pages 263-281
Comparison of Generalized Geometric Programming Algorithms....Pages 283-320
Solving Geometric Programs Using GRG: Results and Comparisons....Pages 321-332
Dual to Primal Conversion in Geometric Programming....Pages 333-342
A Modified Reduced Gradient Method for Dual Posynomial Programming....Pages 343-353
Global Solutions of Mathematical Programs with Intrinsically Concave Functions....Pages 355-373
Interval Arithmetic in Unidimensional Signomial Programming....Pages 375-387
Signomial Dual Kuhn-Tucker Intervals....Pages 389-405
Optimal Design of Pitched Laminated Wood Beams....Pages 407-419
Optimal Design of a Dry-Type Natural-Draft Cooling Tower by Geometric Programming....Pages 421-439
Bibliographical Note on Geometric Programming....Pages 441-453
Back Matter....Pages 455-460
