Treating problems with many variables

Consider a general optimization problem, with several responses yi, j2 . .., ym, for which we construct models based on a set of coded factors xi, X2,. .., x . What can be done to discover the factor levels that will produce the most satisfactory set of responses  

There are several possibilities. If the number of factors, Xi, allows direct visualization of the fitted models, and if the number of responses is not too large, we can overlay the response surfaces — or better yet their contour curves — and locate the best region by visual inspection. That was the procedure followed in the last section. 

on the other hand, our objective is to maximize or minimize a certain response, while keeping the others subject to certain constraints, we can resort to the linear programming methods — or even non-linear ones — commonly used in engineering. 

Finally, if the problem does not fall into either of these two categories, we can attempt to apply the simultaneous optimization methodology proposed by Derringer and Suich (1980), which can be quite usefiil when used with care. 

The Derringer and Suich method is based on the definition of a desirability function for each response, with values restricted to the [0,1] interval. Zero stands for an unacceptable value, while one is assigned to the most desirable value. The nature of the function depends on the objectives of the experiment, as we shall see. 

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The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

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