# Presolve

Presolve is a preprocess of the lp-model. It looks for ways to simplify it. For example it can delete unused variables and restrictions. Substitute fixed variable values by a constant and so on. The result is a new model that is less complex than the original model and likely solves faster.
The result of presolve can be that there are less variables and/or constraints in the presolved model.

Presolve is not active by default. It must specifically being enabled via the API call set_presolve.
Different presolve options are possible.
The more are chosen, the more presolve can be done and the model could me made simpler thus solved faster, but also presolve takes more time. Presolve will most of the time result in a netto time that is faster than without, but can also result in a slower time. So it is up to the user to decide when and which presolve to use on specific models. There is no general presolve option applicable for all models.

A simple example:

    max: x1 + x2 + x3;

    r1: x2 = 2;
    r2: x1 + x2 <= 6;
    r3: x1 - 2 x3 >= 2;
    r4: x2 + 3 x3 <= 5;

Row r1 is a constraint on one variable. Presolve can detect this and convert it to a bound on that variable. For this, presolve of rows must be activated:

    lp_solve model.lp -wlp con -S1 -wafter -presolverow

    /* Objective function */
    max: +x1 +x2 +x3;

    /* Constraints */
    r2: +x1 +x2 <= 6;
    r3: +x1 -2 x3 >= 2;
    r4: +x2 +3 x3 <= 5;

    /* Variable bounds */
    x2 = 2;

And even this model can be presolved further.
When presolve of columns is active, variables can be eliminated from the model.

    lp_solve model.lp -wlp con -S1 -wafter -presolverow -presolvecol

    /* Objective function */
    max: +x1 +x3 +2;

    /* Constraints */
    r3: +x1 -2 x3 >= 2;

    /* Variable bounds */
    x1 <= 4;
    x3 <= 1;

Note the -wafter option. If this option is not specified, the original model is printed. The reason for this is the moment that write_lp is called. The effect of presolve happens when a solve is done. If write_lp is called before solve then the original model is given. If called after solve then the presolved model is.

As previously stated, presolve can result in removal of variables and/or constraints. For some models, presolve can even find the solution of the model.
For constraints that are removed, all information of them are lost. Presolve has removed them from the matrix and cannot retrieve any information of them anymore. The number of rows is decreased by the number of constraints deleted. get_Nrows does not return the original number of rows, but the number of rows in the new model.
Variables that are removed from the matrix result in less columns in the model. get_Ncolumns does not return the original number of columns, but the number of columns in the new model.
However the information of the removed variables is not lost. Only variables that result in a fixed value are removed by presolve and their value is remembered at that time.

Example in C:

    #include <stdio.h>

    #include "lp_lib.h"

    int main(void)
    {
    # if defined ERROR
    #  undef ERROR
    # endif
    # define ERROR() { fprintf(stderr, "Error\n"); exit(1); }
      lprec *lp;

      if ((lp=make_lp(0, 3)) == NULL)
        ERROR();
      set_col_name(lp, 1, "x1");
      set_col_name(lp, 2, "x2");
      set_col_name(lp, 3, "x3");
      set_maxim(lp);
      if (!str_add_constraint(lp, "0 1 0", EQ, 2))
        ERROR();
      set_row_name(lp, 1, "R1");
      if (!str_add_constraint(lp, "1 1 0", LE, 6))
        ERROR();
      set_row_name(lp, 2, "R2");
      if (!str_add_constraint(lp, "1 0 -2", GE, 2))
        ERROR();
      set_row_name(lp, 3, "R3");
      if (!str_add_constraint(lp, "0 1 3", LE, 5))
        ERROR();
      set_row_name(lp, 4, "R4");
      if (!str_set_obj_fn(lp, "1 1 1"))
        ERROR();

      write_LP(lp, stdout);
      set_presolve(lp, PRESOLVE_ROWS + PRESOLVE_COLS, 0);
      solve(lp);
      print_objective(lp);
      print_solution(lp, 1);
      print_constraints(lp, 1);

      print_duals(lp);

      write_LP(lp, stdout);

      delete_lp(lp);

    }

This gives:

    /* Objective function */
    max: +x1 +x2 +x3;

    /* Constraints */
    R1: +x2 = 2;
    R2: +x1 +x2 <= 6;
    R3: +x1 -2 x3 >= 2;
    R4: +x2 +3 x3 <= 5;

    Model name:  '' - run #1
    Objective:   Maximize(R0)

    SUBMITTED
    Model size:        4 constraints,       3 variables,            7 non-zeros.
    Sets:                                   0 GUB,                  0 SOS.

    Presolve O:1 -> Reduced rows:    3, cols:    1 --- changed bnds:    4, Ab:    0.

    PRESOLVE             Elimination loops performed.......... O3:M3:I5
                       1 empty or fixed variables............. REMOVED.
                       3 empty or redundant constraints....... REMOVED.
                       4 bounds............................... TIGHTENED.
                         [           +2 < Z < +7           ]

    REDUCED
    Model size:        1 constraints,       2 variables,            2 non-zeros.
    Sets:                                   0 GUB,                  0 SOS.
    Row-types:         0 LE,                1 GE,                   0 EQ.

    Using DUAL simplex for phase 1 and PRIMAL simplex for phase 2.
    The primal and dual simplex pricing strategy set to 'Devex'.

    Optimal solution                   7 after          0 iter.

    Excellent numeric accuracy ||*|| = 0

     MEMO: lp_solve version 5.5.2.14 for 32 bit OS, with 64 bit REAL variables.
          In the total iteration count 0, 0 (100.0%) were bound flips.
          There were 0 refactorizations, 0 triggered by time and 0 by density.
           ... on average 0.0 major pivots per refactorization.
          The largest [etaPFI v1.4] fact(B) had 0 NZ entries, 0.0x largest basis.
          The constraint matrix inf-norm is 2, with a dynamic range of 2.
          Time to load data was 0.235 seconds, presolve used 0.140 seconds,
           ... 0.047 seconds in simplex solver, in total 0.422 seconds.

    Value of objective function: 7

    Actual values of the variables:
    x1                              4
    x2                              2
    x3                              1

    Actual values of the constraints:
    R3                              2

    Objective function limits:
                                     From            Till       FromValue
    x1                                  0          1e+030         -1e+030
    x3                                  0          1e+030         -1e+030

    Dual values with from - till limits:
                               Dual value            From            Till
    R3                                  0         -1e+030          1e+030
    x1                                  1               4          1e+030
    x3                                  1         -1e+030               1
    /* Objective function */
    max: +x1 +x3 +2;

    /* Constraints */
    R3: +x1 -2 x3 >= 2;

    /* Variable bounds */
    x1 <= 4;
    x3 <= 1;

The result only shows values of contraint R3, the one that is kept. The information of the removed constraints is no longer there.
The variable result however is not lost. x2 is removed by presolve, but its value is still known. On the other hand, note also that the variable does not appear in the sensitivity result.

Almost all API calls return information of the presolved model. get_Nrows and get_Ncolumns return the number of rows and columns of the presolved model. To know the original values, get_Norig_rows and get_Norig_columns must be used:

    printf("get_Nrows: %d\n", get_Nrows(lp));
    printf("get_Norig_rows: %d\n", get_Norig_rows(lp));
    printf("get_Ncolumns: %d\n", get_Ncolumns(lp));
    printf("get_Norig_columns: %d\n", get_Norig_columns(lp));

    get_Nrows: 1
    get_Norig_rows: 4
    get_Ncolumns: 2
    get_Norig_columns: 3

To retrieve the variable data of all variables, including the presolved variables, API call get_var_primalresult must be used. This is the only call available to get all information.
The following code prints the values of all variables:

    int column, Nrows, Ncolumns;

    Ncolumns = get_Norig_columns(lp);
    Nrows = get_Norig_rows(lp);
    for (column = 1; column <= Ncolumns; column++) {
      printf("x%d: %f\n", column, get_var_primalresult(lp, Nrows + column));
    }

With result:

    x1: 4.000000
    x2: 2.000000
    x3: 1.000000

get_var_primalresult also returns information about rows.
The following code prints the results of all rows:

    int row, Nrows;

    Nrows = get_Norig_rows(lp);

    for (row = 1; row <= Nrows; row++) {
      printf("R%d: %f\n", row, get_var_primalresult(lp, row));
    }

With result:

    R1: 0.000000
    R2: 0.000000
    R3: 2.000000
    R4: 0.000000

You could think here that a result for all original rows are returned, but this is not the case. Only the result of constraints that were kept in the model are given. The other rows will return 0!

There is also a possibility to know which rows and columns are removed from the model. get_orig_index can be used for that.
The following code demonstrates it:

    int row, Nrows;
    int column, Ncolumns;

    Ncolumns = get_Ncolumns(lp);
    Nrows = get_Nrows(lp);

    for (row = 1; row <= Nrows; row++)
      printf("row %d was row R%d\n", row, get_orig_index(lp, row));

    for (column = 1; column <= Ncolumns; column++)
      printf("column %d was column x%d\n", column, get_orig_index(lp, Nrows + column));

With result:

    row 1 was row R3
    column 1 was column x1
    column 2 was column x3

So from this it can be determined that only row R3 and columns x1 and x3 are kept in the model. The others are removed by presolve.

Again note that all other API calls return information about the presolved model. So for example get_ptr_primal_solution must be used like this:

    int row, Nrows;
    int column, Ncolumns;
    double *pv;

    get_ptr_primal_solution(lp, &pv);

    Ncolumns = get_Ncolumns(lp);
    Nrows = get_Nrows(lp);

    for (row = 1; row <= Nrows; row++)
      printf("row %d: %f\n", row, pv[row]);

    for (column = 1; column <= Ncolumns; column++)
      printf("column %d: %f\n", column, pv[Nrows + column]);

With result:

    row 1: 2.000000
    column 1: 4.000000
    column 2: 1.000000

Row and column names of the original model can be obtained via get_origrow_name and get_origcol_name:

    int row, Nrows;
    int column, Ncolumns;

    Ncolumns = get_Norig_columns(lp);
    Nrows = get_Norig_rows(lp);

    for (row = 1; row <= Nrows; row++)
     printf("row %d: %s\n", row, get_origrow_name(lp, row));

    for (column = 1; column <= Ncolumns; column++)
     printf("column %d: %s\n", column, get_origcol_name(lp, column));

With result:

    row 1: R1
    row 2: R2
    row 3: R3
    row 4: R4
    column 1: x1
    column 2: x2
    column 3: x3

Row and column names of the presolved model can be obtained via get_row_name and get_col_name:

    int row, Nrows;
    int column, Ncolumns;

    Ncolumns = get_Ncolumns(lp);
    Nrows = get_Nrows(lp);

    for (row = 1; row <= Nrows; row++)
     printf("row %d: %s\n", row, get_row_name(lp, row));

    for (column = 1; column <= Ncolumns; column++)
     printf("column %d: %s\n", column, get_col_name(lp, column));

With result:

    row 1: R3
    column 1: x1
    column 2: x3