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

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 brief article within the lawsuits 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 which could usually be expressed because the challenge of minimizing a numerical fee functionality, termed a "generalized polynomial," which include a sum of phrases, the place each one time period is a manufactured from a favorable consistent and the layout variables, raised to arbitrary powers. He saw that if the variety of phrases exceeds the variety of variables by way of one, the optimum values of the layout variables might be simply discovered through fixing a suite of linear equations. moreover, sure invariances of the relative contribution of every time period to the whole rate should be deduced. The mathematical intricacies in Zener's strategy quickly raised the interest of R. J. Duffin, the prestigious mathematician from Carnegie Mellon collage 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 suggestions in such difficulties have been conducted by means of Duffin and Zener via inequalities, instead of the extra universal strategy of the Kuhn-Tucker theory.

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A detailed analysis of this rather broad class of optimization problems can be found in Peterson and Ecker (Ref. 18) and the references cited therein. 3. Our third example comes from the optimal location of a new facility relative to existing facilities. We suppose that there are Ie existing facilities with fixed locations b\ b 2 , ••• , b" in Em, and we assume E. L. PetersoR that for each facility i there is a cost d,Cj, b') of choosing the new facility location I relative to bi. In many instances, the functions d, are just "metrics" that reflect the cost of shipping material between the two locations.

2. If w(z)~ TJ-l L i=l p;llzi-bdP'+z,,-bTJ and W~ETJ' Pi> 1, i = 1, 2, ... , 1/ -1, then TJ-l = L sup i=l [(iZi-P;llzi-biIP,]+ sup [(TJZTJ-(ZTJ-b,,)], ZiEEl Z"eEl which is clearly finite if and only if (TJ differential calculus shows that w(t) = = 1, in which case an application of the TJ-l L (qi11(d i=l q, + bi(i) + b TJ , 51 Geometric Programming where q; is determined from p; by the equation pi 1 +qi 1 Ct/ (t) = ,,-1 L (q i 1 1(;lq; + b;{;) + b" ;=1 and = 1. Consequently, fi = {t E E" 1(" = I}.

Consequently, the P optimality conditions frequently characterize the optimal solution set g* for Problem d. 3. 2. The Conjugate Transformation, Subgradients, and Convex Analysis. The conjugate transformation evolved from the classical Legendre transformation but was first studied in great detail only rather recently by Fenchel (Refs. 31 and 32). [For a very thorough and modern treatment of both transformations see the recent book by Rockafellar (Ref. ] We now briefly describe only those of their properties that are relevant to geometric programming.