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

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 brief 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 which can frequently be expressed because the challenge of minimizing a numerical rate functionality, termed a "generalized polynomial," which includes 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 way of one, the optimum values of the layout variables might be simply discovered by means of fixing a suite of linear equations. additionally, definite invariances of the relative contribution of every time period to the full expense may be deduced. The mathematical intricacies in Zener's approach quickly raised the interest of R. J. Duffin, the celebrated 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 houses of the optimum options in such difficulties have been conducted by means of Duffin and Zener using inequalities, instead of the extra universal technique of the Kuhn-Tucker theory.

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1 and 2) and Duffin (Refs. 3 and 4), geometric programming has undergone rapid development, especially with Reprinted with permission from SIAM Review, Vol. 18. Copyright © 1976, Society for Industrial and Applied Mathematics. All rights reserved. 2 Department of Mathematics and Graduate Program in Operations Research, North Carolina State University, Raleigh 27609. 1 31 32 E. L. Petenon the appearance of the first book on the subject by Duffin, Peterson, and Zener (Ref. 5). Although its essence and scope have recently been broadened and amplified by Peterson (Refs.

For purposes of easy reference and mathematical precision, the resulting geometric programming Problem A is now given the following formal definition in terms of classical terminology and notation. Problem A. C ~{(x, Consider the objective function G whose domain IC) IXk E Ck> k E {OJ ul, and (xi, Kj) E ct, j E J}, and whose functional value where ct ~{(xj, Kj)leither Kj =0 and sup (x j, d j)< +00, or Kj>O and xj E KjCJ, dIeD; and SUP (xi, di ) if Kj = 0 and sup (xi, di ) < +00, . d'eD/ gt (xi, Kj) ~ { dieDi Kjgi(X I I Kj) if Kj > 0 and xj E KPi' Using the feasible solution set S ~{(x, K)E C IXEX, and gi(Xi)~O, i EI}, calculate both the problem infimum qJ ~ inf G(x, K) (X,K)eS and the optimal solution set S* ~{(x, K) E S IG(x, K) = qJ}.

FA = <1>(0). 2. ;(,fA. Proof. Consider the following pair of dual programs (7), (8): min {Go(xo) IGdXk) ~ E (k = 1, 2, ... ,m), x max {V(y, A)-EA I(y, A) E dom V, y E E ~}, gil}, (11) (12) where If (5) is subconsistent, then for arbitrary E > 0, (11) is strictly consistent. (0) is finite for each 0 < E:S; i. Bamala which implies AlA ~MB and consistency of (6). Suppose that AlA

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