In the MPS standard, the constraint matrix, the matrix of linear elements, is input by columns; firstly the names of the constraints are specified in the ROWS section and then variable names and the corresponding matrix coefficients are set one at a time in the COLUMNS section. While there is some justification for this form of matrix entry for linear programming problems -- the principal solution algorithm for such problems, the simplex method , is usually column oriented -- there seems no good reason why the coefficients of linear elements might not also be input by rows. After all, it is more natural to think of specifying the constraints for a problem one at a time. Furthermore, requiring that a complete row or column has been specified before the next may be processed is unnecessarily restrictive.
We thus allow the data to be input in a either a group-wise (row-wise) or variable-wise (column-wise) fashion. In a group/row-wise scheme, one or two coefficients and their variable/column names are specified for a given group/row; for a variable/column-wise scheme, one or two coefficients and their group/row names are specified for a given variable/column. We do, however, still require that in a group/row-wise storage scheme, the names of all the variables/rows which appear in linear elements are completely specified before the coefficients are input. Similarly, in a variable/column-wise storage scheme, the names of all the groups/rows which have a linear element must be completely specified before the coefficients are input. This allows for some checking of the input data.
If the groups/rows are specified first, there is no requirement that variables/columns are input one at a time (but of course they may be). When processing the data file, variable/column names should be inspected to see if they are new or where they have appeared before. Likewise, if the variables/columns are specified first, there is no requirement that groups/rows are ordered on input. The coordinates of new data values can then be stored as a linked triple (group/row, variable/column, value). Conversion from such a component-wise input scheme to a row or column based storage scheme may be performed very efficiently if desired (see [5, pp30-31], and subroutine MC39 in the Harwell Subroutine Library).
If a variable/column-wise input scheme is to be adopted, the data file will contain a GROUPS/ROWS/CONSTRAINTS indicator card and section followed by a VARIABLES/COLUMNS card and section. The allowed data cards are discussed in Section 3.2.6 and Section 3.2.7. If a group/row-wise input scheme is to be adopted, the data file will contain a VARIABLES/COLUMNS indicator card and section followed by a GROUPS/ROWS/CONSTRAINTS card and section. The data cards for this scheme are discussed in Section 3.2.8 and Section 3.2.9.