Read e-book online Advances in Geometric Programming PDF

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 very important contributions to physics and engineering, released a quick article within the complaints of the nationwide Academy of Sciences entitled" A Mathe­ matical reduction in Optimizing Engineering layout. " listed here Zener thought of the matter of discovering an optimum engineering layout that could frequently be expressed because the challenge of minimizing a numerical fee functionality, termed a "generalized polynomial," such as a sum of phrases, the place every one time period is a made 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 by means of one, the optimum values of the layout variables could be simply came upon by way of fixing a collection of linear equations. moreover, sure invariances of the relative contribution of every time period to the whole fee could be deduced. The mathematical intricacies in Zener's procedure 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 houses of the optimum suggestions in such difficulties have been performed by way of Duffin and Zener because of inequalities, instead of the extra universal strategy of the Kuhn-Tucker theory.

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2. Our second example comes from the minimization of quadratic functions where H is an arbitrary constant matrix and h is an arbitrary constant vector. A factorization of the coefficient matrix H (which is, of course, associated with quadratic nonlinearities) is the key to effectively applying geometric programming to quadratic programming. More specifically, linear algebra is used to compute matrices D and gg such that H =DTD _ggTgg, where T indicates the transpose operation. In terms of D and gg the 35 Geometric Programming quadratic function Q(i) = @(Di, Di) -(g}Ji, g}Ji») + (h, i).

6. First, we make the following choices: i={1,2, ... '" no+1=ml~nl' np_l+1=mp~np=n, X £ column space of [a qr ], where [a qr ] is any n x m matrix. s4 are completely separable. To relate problem A to constrained signomiai optimization, explicitly eliminate the vector space condition x E X by the (essentially linear) transformation m Xq = L aqr log tN q = 1, 2, ... , n. ,. ,p, and t>O. Of course, the preceding procedure is usually reversed in practice; that is, the signomial form of Problem A tends to occur more often than Problem 43 Geometric Programming A in real-world applications, but is transformed into problem A so that the complete separability of the resulting exponential functions can be exploited.

J? J? Now, when g is convex and closed, the definition of L, and the symmetry of the conjugate transformation clearly imply that the preceding minimization problem is essentially Problem d. Consequently, it is not unnatural to consider the preceding maximization problem simultaneously with Problem d and term it the "geometric dual problem" 00. Problem M. J? J? :r I~(AI') = 'it}. :r have trivial solutions. J?

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

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