Simplification Linearization Objective function

Remark 1 The main motivation behind the development of the simplified superstructure was to end up with a mathematical model that features only linear constraints while the nonlinearities appear only in the objective function. Yee et al. (1990a) identified the assumption of isothermal mixing which eliminates the need for the energy balances, which are the nonconvex, nonlinear equality constraints, and which at the same time reduces the size of the mathematical model. These benefits of the isothermal mixing assumption are, however, accompanied by the drawback of eliminating from consideration a number of HEN structures. Nevertheless, as has been illustrated by Yee and Grossmann (1990), despite this simplification, good HEN structures can be obtained. 

A further simplification can-be made to the Bayes classifier if the covariance matrices for both groups are known to be or assumed to be similar. This condition implies that the correlations between variables are independent of the group to which the objects belong. Extreme examples are illustrated in Figure 5. In such cases the groups are linearly separable and a linear discriminant function can be evaluated. 

A concrete example of the variational principle is provided by the Hartree-Fock approximation. This method asserts that the electrons can be treated independently, and that the -electron wavefimction of the atom or molecule can be written as a Slater determinant made up of orbitals. These orbitals are defined to be those which minimize the expectation value of the energy. Since the general mathematical form of these orbitals is not known (especially in molecules), then the resulting problem is highly nonlinear and formidably difficult to solve. However, as mentioned in subsection (A 1.1.3.2). a common approach is to assume that the orbitals can be written as linear combinations of one-electron basis functions. If the basis functions are fixed, then the optimization problem reduces to that of finding the best set of coefficients for each orbital. This tremendous simplification provided a revolutionary advance for the application of the Hartree-Fock method to molecules, and was originally proposed by Roothaan in 1951. A similar form of the trial function occurs when it is assumed that the exact (as opposed to Hartree-Fock) wavefimction can be written as a linear combination of Slater determinants (see equation (A 1.1.104 ) ). In the conceptually simpler latter case, the objective is to minimize an expression of the form... 

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