2.1 Problem, Elemental and Internal Variables

A nonlinear element function $f_j$ is assumed to be a function of the problem variables $\bar{x}_j$, a subset of the overall variables $x$. Suppose that $\bar{x}_j$ has $n_j$ components. Then one can consider the nonlinear element function to be of the structural form $f_j (v_1,\cdots,v_{n_j})$, where we assign $v_1 = \bar{x}_{j1},\cdots, v_{n_j} = \bar{x}_{jn_{j}}$. The elemental variables for the element function $f_j$ are the variables $v$ and, while we need to associate the particular values $\bar{x}_j$ with $v$, it is the elemental variables which are important in defining the character of the nonlinear element functions.

As an example, the first nonlinear element function for a particular problem might be

$$(x_{29} + x_3 - 2x_{17})e^{x_{29} - x_{17}}$$ (2.5)

which has the structural form

$$f_1 (v_1, v_2, v_3) = (v_1 + v_2 - 2v_3) e^{v_{1} - v_{3}},$$ (2.6)

where we need to assign $v_1 = x_{29}, v_2 = x_3$ and $v_3 = x_{17}$. For this example, there are three elemental variables.

The example may be used to illustrate a further point. Although $f_1$ is a function of three variables, the function itself is really only composed of two independent parts; the product of $v_1 + v_2 - 2 v_3$ with $e^{v_1 - v_3}$, or, if we write $u_1 = v_1 + v_2 - 2 v_3$ and $u_2 = v_1 - v_3$, the product of $u_1$ with $e^{u_2}$. The variables $u_1$ and $u_2$ are known as internal variables for the element function. They are obtained as linear combinations of the elemental variables. The important feature as far as an optimization procedure is concerned is that each nonlinear function involves as few internal variables as possible, as this allows for compact storage and more efficient derivative approximation.

It frequently happens, however, that a function does not have useful internal variables. For instance, another element function might have structural form

$$f_2 (v_1, v_2) = v_1 \sin v_2,$$ (2.7)

where for example $v_1 = x_6$ and $v_2 = x_{12}$. Here, we have broken $f_2$ down into as few pieces as possible. Although there are internal variables, $u_1 = v_1$ and $u_2 = v_2$, they are the same in this case as the elemental variables and there is no virtue in exploiting them. Moreover it can happen that although there are special internal variables, there are just as many internal as elemental variables and it therefore doesn't particularly help to exploit them. For instance, if

$$f_3 (v_1, v_2) = (v_1 + v_2) \log (v_1 - v_2),$$ (2.8)

where, for example, $v_1 = x_{12}$ and $v_2 = x_2$, the function can be formed as $u_1 \log u_2$, where $u_1 = v_1 + v_2$ and $u_2 = v_1- v_2$. But as there are just as many internal variables as elementals, it will not normally be advantageous to use this internal representation. Finally, although an element function may have useful internal variables, the user may decide not to exploit them. The optimization procedure should still work but at the expense of extra storage and computational effort.

In general, there will be a linear transformation from the elemental variables to the internal ones. For example in (2.6), we have

$$\left( \begin{array}{c} u_1 \\ u_2 \end{array} \right) = ... ...\left( \begin{array}{c} v_1 \\ v_2\\ v_3 \end{array} \right)$$ (2.9)

while in (2.7), we have

$$\left( \begin{array}{c} u_1 \\ u_2 \end{array} \right) = ... ...ight) \left( \begin{array}{c} v_1 \\ v_2 \end{array} \right)$$ (2.10)

In general the transformation will be of the form

$$u = Wv$$ (2.11)

and this transformation is useful if the matrix $W$ has fewer rows than columns.