3.2.6 The GROUPS, ROWS or CONSTRAINTS Data Cards
(variable/column-wise)

The GROUPS, ROWS and CONSTRAINTS indicator cards are used interchangeably to announce the names of the groups which make up the objective function or, for constrained problems, the names of the constraints (or rows, as they are often known in linear programming applications). The user may give a scaling factor for the groups or constraints. In addition, groups which are linear combinations of previous groups may be specified. The syntax for the data cards which follow these indicator cards is given in Figure 3.7.

Figure 3.7: Possible data cards for GROUPS, ROWS or CONSTRAINTS
(column-wise)

The one- or two-character string in data field 1 specifies the type of group, row or constraint to be input. Possible values for the first character are:

N :

the group is to be specially marked (for constrained problems, the group/row is an objective function group/row).

G :

the group is to use an extra ``artificial'' variable; this variable will only occur in this particular group, will be non-negative and its value will be subtracted from the group function. For constrained problems, this is equivalent to requiring the constraint/row be non-negative; the extra variable is then a surplus variable and whether it is used explicitly (considered as a problem variable) or implicitly will depend upon the optimization technique to be used. Thus, if the problem variables are $x$, and the $k$-th group has a linear element $a_k^T x - b_k$, the linear element that will be passed to the optimization procedure could be $a_k^T x - y_k - b_k$, for some non-negative variable $y_k$.

L :

the group is to use an extra ``artificial'' variable; this variable will only occur in this particular group, will be non-negative and its value will be added to the group function. For constrained problems, this is equivalent to requiring the constraint/row be non-positive; the extra variable is then a slack variable and may be used explicitly or implicitly by the optimization procedure. Thus, if the linear element is as specified above, the linear element that will be passed to the optimization procedure could be $a_k^T x + y_k - b_k$, for some non-negative variable $y_k$.

E :

the group is a normal one (for constrained problems, the row/constraint is an equality),

X and Z :

an array of groups are to be defined at once. When the first character is an X or Z, the second character may be one of N, G, L or E. The resulting array of groups then each has the characteristics of an N, G, L or E group as just described.

D :

the group is to be formed as a linear combination of two previous groups. When the first character is a D, the second character may be one of N, G, L or E. The resulting group then has the characteristics of an N, G, L or E group as just described.

The string group-name in data field 2 gives the name of the group (or row or constraint) under consideration. This name may be up to ten characters long, excepting that the name `SCALE' is not allowed. For X data cards, the expanded array name must be valid and the integer indices must have been defined in a parameter assignment (see Section 3.2.3).

The string $\$\$\$\$\$\$\$\$\$ in data field 3 may be blank; this happens when field 2 is merely announcing the name of a group. If it is not blank, it is used for two purposes.

• It may be used to announce that the group function under consideration is to be scaled, that is divided by a constant scale factor; in this case field 3 will contain the string `SCALE'. If the first character in field 1 is a Z, the string in data field 5 gives the name of a previously defined real parameter and the numerical value associated with this parameter gives the scale factor. Otherwise, the string numerical-vl, occupying up to 12 locations in data field 4, contains the scale factor. Fields 5 and 6 are not then used.
• If the first character in field 1 is a D, the current group is to be formed as a linear combination of the groups mentioned in fields 3 and 5; the multiplication factors are then recorded in fields 4 and 6 respectively. Thus we will have
  
  $$\mbox{group in field~2} = \mbox{group in field~3} * \mbox{field~4} + \mbox{group in field~5} * \mbox{field~6}.$$
  
  
  In this case, the names of the groups in fields 3 and 5 must have already been defined. The multiplication factors may occupy up to 12 locations in fields 4 and 6.