2.2 Element and Group Types

It is quite common for large nonlinear programming problems to be defined in terms of many nonlinear elements. It is also common that these elements, although using different problem variables, are structurally the same as each other. For instance, the function

$$\sum_{i = 1}^{n-1} (x_i x_{i + 1})^i$$ (2.12)

naturally decomposes into the sum of $n-1$ group functions, $\alpha, \alpha^2, \cdots , \alpha^{n-1}$. Each group is a nonlinear element function $v_1 v_2$ of the two elemental variables $v_1$ and $v_ 2$ evaluated for different pairs of problem variables. More commonly, the elements may be arranged into a few classes; the elements within each class are structurally the same. For example, the function

$$\sum_{i = 1}^{n-1} (x_i x_{i + 1} + x_1/x_i)^i$$ (2.13)

naturally decomposes into the sum of the same $n-1$ group functions. Each group is the sum of two nonlinear elements $v_1 v_2$ (where $v_1 = x_i$ and $v_2 = x_{i+1}$) and $v_1 / v_2$ (where $v_1 = x_1$ and $v_2 = x_{i}$). A further common occurrence is the presence of elements which have the same structure, but which differ in using different problem variables and other auxiliary parameters. For instance, the function

$$\sum_{i = 1}^{n-1} (ix_i x_{i + 1})^i$$ (2.14)

naturally decomposes into the sum of the same $n-1$ group functions. Each group is a nonlinear element $p_1 v_1 v_2$ of the single parameter $p_1$ and two elemental variables $v_1$ and $v_2$ evaluated for different values of the parameter and pairs of problem variables. Any two elements which are structurally the same are said to be of the same type. Thus examples (2.12) and (2.14) use a single element type, where as (2.13) uses two types. When defining the data for problems of the form (2.1)-(2.4), it is unnecessary to define each nonlinear element in detail. All that is actually needed is to specify the characteristics of the element types and then to identify each $f_j$ by its type and the indices of its problem variables and (possibly) auxiliary parameters.

The same principal may be applied to group functions. For example, the group functions that make up

$$\sum_{i = 1}^{n-1} (x_i x_{i + 1})^2$$ (2.15)

have different arguments but are structurally all the same, each being of the form $g_i ( \alpha ) = \alpha^2$. As a slightly more general example, the group functions for

$$\sum_{i = 1}^{n-1} i(x_i x_{i + 1})^2$$ (2.16)

have different arguments and depend upon different values of a parameter but are still structurally all the same, each being of the form $g ( \alpha ) = p_1 \alpha^2$ for some parameter $p_1$. Any two group functions which are structurally the same are said to be of the same type; the structural function is known as the group type and its argument is the group-type variable. Once again, using group types makes the task of specifying the characteristics of individual group functions more straightforward. The group type $g(\alpha ) = \alpha$ is known as the trivial type. Trivial groups occur very frequently and are considered to be the default type. It is then only necessary to specify non-trivial group types.