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Download PDF by M. Avriel (auth.), Mordecai Avriel (eds.): Advances in Geometric Programming

By M. Avriel (auth.), Mordecai Avriel (eds.)

ISBN-10: 1461582857

ISBN-13: 9781461582854

ISBN-10: 1461582873

ISBN-13: 9781461582878

In 1961, C. Zener, then Director of technology at Westinghouse Corpora­ tion, and a member of the U. S. nationwide Academy of Sciences who has made vital contributions to physics and engineering, released a brief article within the complaints of the nationwide Academy of Sciences entitled" A Mathe­ matical relief in Optimizing Engineering layout. " listed here Zener thought of the matter of discovering an optimum engineering layout which may frequently be expressed because the challenge of minimizing a numerical expense functionality, termed a "generalized polynomial," such as a sum of phrases, the place each one time period is a manufactured from a good consistent and the layout variables, raised to arbitrary powers. He saw that if the variety of phrases exceeds the variety of variables via one, the optimum values of the layout variables might be simply stumbled on by means of fixing a collection of linear equations. in addition, sure invariances of the relative contribution of every time period to the entire fee could be deduced. The mathematical intricacies in Zener's approach quickly raised the interest of R. J. Duffin, the prestigious mathematician from Carnegie­ Mellon college who joined forces with Zener in laying the rigorous mathematical foundations of optimizing generalized polynomials. Interes­ tingly, the research of optimality stipulations and homes of the optimum ideas in such difficulties have been conducted by means of Duffin and Zener by using inequalities, instead of the extra universal method of the Kuhn-Tucker theory.

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Extra info for Advances in Geometric Programming

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Q <0). Then g(X)={(Ixf)l/P -0 ,L... If x 2:0,x)s-0=-rg*(y) = h' -

Consequently, the conjugate transform Ct/ exists when w has at least one subgradient t, a condition that is known to be fulfilled when w is convex. Actually, the conjugate transform Ct/ restricted (in the set-theoretic sense) to the domain UZEWdW(Z) is termed the "Legendre transform" of wand has been a major tool in the study of classical mechanics, thermodynamics, and differential equations [as described, for example, by Courant and Hilbert (Ref. 34)]. Usually, the domain fi of the conjugate transform Ct/ consists of both UZEWdW(Z) and some of its limit points.

5), and since the theory for the unconstrained case is far simpler than that for the constrained case, we initially limit our attention to the unconstrained case. 2. 1. The Unconstrained Case. that is, Let uy be the "dual" of the cone a;, uy ~ {,~ E En I0 ~ (x, y) for each x E a;}. Rather elementary considerations show that uy is generally a closed convex cone. , "finitely generated") when a; is polyhedral; and uy is the "orthogonal complement" a;1- of a; when a; is actually a vector space. 1. Some of the following subsections can be omitted.

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Advances in Geometric Programming by M. Avriel (auth.), Mordecai Avriel (eds.)

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