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

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 vital contributions to physics and engineering, released a quick article within the court cases 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 may usually be expressed because the challenge of minimizing a numerical price functionality, termed a "generalized polynomial," along with a sum of phrases, the place each one time period is a made of 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 way of one, the optimum values of the layout variables could be simply chanced on via fixing a collection of linear equations. in addition, sure invariances of the relative contribution of every time period to the complete rate will be deduced. The mathematical intricacies in Zener's technique quickly raised the interest of R. J. Duffin, the celebrated 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 through Duffin and Zener via inequalities, instead of the extra universal strategy of the Kuhn-Tucker theory.

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6. ,qk 0, L Yi qk ~ 1, h otherwise, M. Hamala 30 and we have the following dual pair: max {L (Xi + di)Po IL(Xi + d 10 min {L diYi - i )Pk Ik ~ b/k (k = 1, 2, ... , YE qg} 10 r. References 1. DUFFIN, R. , PETERSON, E. , Geometric Programming, John Wiley and Sons, New York, 1967. 2. ROCKAFELLAR, R. , Nonlinear Programming, talk given at the American Mathematical Society Summer Seminar on the Mathematics of the Decision Sciences, Stanford University, 1967.

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. (l.

EGGLESTON, H. , Convexity, Cambridge University Press, New York, 1958. 4. , Mathematical Methods and Theory in Games, Programming and Economics, Vol. , Addison-Wesley Publishing Company, Reading, Massachusetts, 1959. 5. VALENTINE, F. , Convex Sets, McGraw-Hill, New York, 1964. 2 Geometric Programming1 E. L. PETERSON 2 Abstract. Contrary to popular belief, geometric programming is not just a special technique for studying the very important class of posynomial (optimization) problems. It is really a very general mathematical theory that is especially useful for studying a large class of separable problems.

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