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 technological know-how 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 which can usually be expressed because the challenge of minimizing a numerical price 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 way of one, the optimum values of the layout variables may be simply discovered via fixing a collection of linear equations. in addition, yes invariances of the relative contribution of every time period to the whole expense could be deduced. The mathematical intricacies in Zener's strategy 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 strategies in such difficulties have been conducted through Duffin and Zener using inequalities, instead of the extra universal method of the Kuhn-Tucker theory.

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There is, however, an important class of functions whose gradients are also subgradients-the class of convex functions. 1. To relate the conjugate transform to subgradients, observe that if t E dW (z), then (t, z') - w (z') ~ (t, z) - w (z) for each z' E W, which in turn clearly implies that t E fi and that Ct/ (t) = -[ w (z) + (t, -z)]. Hence, Ct/ (t) is simply the negative of the intercept of the corresponding supporting hyperplane with the w' axis. 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.

It is really a very general mathematical theory that is especially useful for studying a large class of separable problems. Its practical efficacy is due mainly to the fact that many important (seemingly inseparable) problems can actually be formulated as separable geometric programming problems, by fully exploiting their linear algebraic structure. Some examples are: nonlinear network flow problems (both single-commodity and multicommodity), discrete optimal control problems with linear dynamics, optimal location problems of the generalized Fermat type, (lp constrained) Ip regression problems, chemical equilibrium problems, ordinary programming problems, (quadratically constrained) quadratic programming problems, and general algebraic programming problems.

The following example indicates the generality of geometric programming. 8. First, make the following choices: J=0, nk =m and Ck £ Co, kE{O}uI, where Co is an arbitrary subset of Em, where there is a total of 1 + 0(1) identity matrices U that are m x m. To relate Problem A to ordinary programming, explicitly eliminate the vector space condition x E X by the linear transformation Problem A is then clearly equivalent to the following very general "ordinary programming problem": Minimize subject to go(z) gi(Z)~O, i EI, 4S Geometric Programming Thus ordinary programming can be viewed as a special case of geometric programming.

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