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.

Klappentext

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.

Inhalt

1. Geometric Programming in Terms of Conjugate Functions.- 2. Geometric Programming.- 3. Optimality Conditions in Generalized Geometric Programming.- 4. Saddle Points and Duality in Generalized Geometric Programming.- 5. Constrained Duality via Unconstrained Duality in Generalized Geometric Programming.- 6. Fenchel's Duality Theorem in Generalized Geometric Programming.- 7. Generalized Geometric Programming Applied to Problems of Optimal Control: I. Theory.- 8. Projection and Restriction Methods in Geometric Programming and Related Problems.- 9. Transcendental Geometric Programs.- 10. Solution of Generalized Geometric Programs.- 11. Current State of the Art of Algorithms and Computer Software for Geometric Programming.- 12. A Comparison of Computational Strategies for Geometric Programs.- 13. Comparison of Generalized Geometric Programming Algorithms.- 14. Solving Geometric Programs Using GRG: Results and Comparisons.- 15. Dual to Primal Conversion in Geometric Programming.- 16. A Modified Reduced Gradient Method for Dual Posynomial Programming.- 17. Global Solutions of Mathematical Programs with Intrinsically Concave Functions.- 18. Interval Arithmetic in Unidimensional Signomial Programming.- 19. Signomial Dual KuhnTucker Intervals.- 20. Optimal Design of Pitched Laminated Wood Beams.- 21. Optimal Design of a Dry-Type Natural-Draft Cooling Tower by Geometric Programming.- 22. Bibliographical Note on Geometric Programming.