MPS file format

The main things to know about MPS format are that it is column oriented
(as opposed to entering the model as equations), and everything
(variables, rows, etc.) gets a name.

MPS is an old format, so it is set up as though you were using punch
cards, and is not free format. Fields start in column 2, 5, 15, 25, 40
and 50.  Sections of an MPS file are marked by so-called header cards,
which are distinguished by their starting in column 1.  Although it is
typical to use upper-case throughout the file (like I said, MPS has
long historical roots), many MPS-readers will accept mixed-case for
anything except the header cards, and some allow mixed-case anywhere.
The names that you choose for the individual entities (constraints or
variables) are not important to the solver; you should pick names that
are meaningful to you, or will be easy for a post-processing code to

Here is a little sample model written in MPS format (explained in more
detail below):

 L  LIM1
 G  LIM2
    XONE      COST                 1   LIM1                 1
    XONE      LIM2                 1
    YTWO      COST                 4   LIM1                 1
    YTWO      MYEQN               -1
    ZTHREE    COST                 9   LIM2                 1
    ZTHREE    MYEQN                1
    RHS1      LIM1                 5   LIM2                10
    RHS1      MYEQN                7
 UP BND1      XONE                 4
 LO BND1      YTWO                -1
 UP BND1      YTWO                 1

For comparison, here is the same model written out in lp-format:

min: +XONE +4 YTWO +9 ZTHREE;
LIM1: +XONE +YTWO <= 5;
LIM2: +XONE +ZTHREE >= 10;
XONE <= 4;
YTWO >= -1;
YTWO <= 1;

Strangely, there is nothing in MPS format that specifies the direction
of optimisation.  And there really is no standard "default" direction;
some LP codes will maximize if you don't specify otherwise, others will
minimize, and still others put safety first and have no default and
require you to specify it somewhere in a control program or by a
calling parameter.  If you have a model formulated for minimization
and the code you are using insists on maximization (or vice versa), it
may be easy to convert: just multiply all the coefficients in your
objective function by (-1).  The optimal value of the objective
function will then be the negative of the true value, but the values of
the variables themselves will be correct.

Any line with an asterisk (*) in Column 1 is treated as a comment.

The eight character names used to specify variables, constraints and
other entities are fixed format. Names are not automatically justified,
so blanks are treated just like other characters. For example "ROW1    "
is not the same as " ROW1   ". (Note that some optimisers do not permit
blanks in names.) No case conversion is performed, so "row1    " is
different from "ROW1    ".

Floating point numbers may be specified in free format within the 12
character field (including embedded blanks). The following list describes
the possible ways of writing a number.


        + or -      optional sign character (no sign indicates a positive number)

        digits      optional integer part of the mantissa

        .           optional decimal point (if not present, a decimal point will
                    be assumed after the mantissa digit)

        digits      optional fraction part of the mantissa -the mantissa must
                    contain at least one digit

Exponent (optional):

        D or E      exponent leader

        + or -      optional exponent sign

        digits      exponent digits

Numbers with an absolute value greater than 1010 or less than 10-10 are rejected.

The NAME card can have anything you want, starting in column 15.  The
ROWS section defines the names of all the constraints; entries in
column 2 or 3 are E for equality rows, L for less-than ( <= ) rows, G
for greater-than ( >= ) rows, and N for non-constraining rows (the
first of which would be interpreted as the objective function).  The
order of the rows named in this section is unimportant.

The largest part of the file is in the COLUMNS section, which is the
place where the entries of the A-matrix are put. All entries for a given
column must be placed consecutively, although within a column the
order of the entries (rows) is irrelevant. Rows not mentioned for a
column are implied to have a coefficient of zero.

The RHS section allows one or more right-hand-side vectors to be
defined; most people don't bother having more than one.  In the above
example, the name of the RHS vector is RHS1, and has non-zero values
in all 3 of the constraint rows of the problem.  Rows not mentioned in
an RHS vector would be assumed to have a right-hand-side of zero.

The optional BOUNDS section lets you put lower and upper bounds on
individual variables (no * wild cards, unfortunately), instead of
having to define extra rows in the matrix.  All the bounds that have
a given name in column 5 are taken together as a set.  Variables not
mentioned in a given BOUNDS set are taken to be non-negative (lower
bound zero, no upper bound).  A bound of type UP means an upper bound
is applied to the variable.  A bound of type LO means a lower bound is
applied.  A bound type of FX ("fixed") means that the variable has
upper and lower bounds equal to a single value.  A bound type of FR
("free") means the variable has neither lower nor upper bounds.

There is another optional section called RANGES that I won't go into
here. The final card must be ENDATA, and yes, it is spelled funny.


MPS input format was originally introduced by IBM to express linear
and integer programs in a standard way.  The format is a fixed column
format, so care must be taken that all information is placed in the
correct columns as described below.

The following is not intended as a complete description of MPS format,
but only as a brief introduction.  For more information, the reader is
directed to:

  "Advanced Linear Programming," by Bruce A. Murtagh
  "Computer Solutions of Linear Programs," by J.L. Nazareth

It may be useful to look at an example MPS file while reading this
MPS information.

The following template is a guide for the use of MPS format:

Field:    1           2          3         4         5         6
Columns:  2-3        5-12      15-22     25-36     40-47     50-61

          NAME   problem name


           type     name

                   column       row       value     row      value
                    name        name                name
                    rhs         row       value     row      value
                    name        name                name
                    range       row       value     row      value
                    name        name                name

           type     bound       column    value
                    name        name

           type     CaseName    SOSName   SOSpriority
                    CaseName    VarName1  VarWeight1
                    CaseName    VarName2  VarWeight2

                    CaseName    VarNameN  VarWeightN



A. In the ROWS section, each row of the constraint matrix must have a
   row type and a row name specified.  The code for indicating row type
   is as follows:

                     type      meaning
                      E    equality
                      L    less than or equal
                      G    greater than or equal
                      N    objective
                      N    no restriction

B. In the COLUMNS section, the names of the variables are defined along
   with the coefficients of the objective and all the nonzero constraint
   matrix elements.  It is not necessary to specify columns for slack or
   surplus variables as this is taken care of automatically.

C. The RHS section contains information for the right-hand side of the problem.

D. The RANGES section is for constraints of the form:  h <= constraint <= u .
   The range of the constraint is  r = u - h .  The value of r is specified
   in the RANGES section, and the value of u or h is specified in the RHS
   section.  If b is the value entered in the RHS section, and r is the
   value entered in the RANGES section, then u and h are thus defined:

        row type       sign of r       h          u
           G            + or -         b        b + |r|
           L            + or -       b - |r|      b
           E              +            b        b + |r|
           E              -          b - |r|      b

E. In the BOUNDS section, bounds on the variables are specified.  When
   bounds are not indicated, the default bounds ( 0 <= x < infinity )
   are assumed.  The code for indicating bound type is as follows:

                    type            meaning
                     LO    lower bound        b <= x (< +inf)
                     UP    upper bound        (0 <=) x <= b
                     FX    fixed variable     x = b
                     FR    free variable      -inf < x < +inf
                     MI    lower bound -inf   -inf < x (<= 0)
                     PL    upper bound +inf   (0 <=) x < +inf
                     BV    binary variable    x = 0 or 1
                     LI    integer variable   b <= x (< +inf)
                     UI    integer variable   (0 <=) x <= b
                     SC    semi-cont variable x = 0 or l <= x <= b
                           l is the lower bound on the variable
                           If none set then defaults to 1

F. Sections RANGES and BOUNDS are optional as are the fields 5 and 6.
   Everything else is required.  In regards to fields 5 and 6, consider
   the following 2 constraints:

                       const1:  2x + 3y <= 6
                       const2:  5x + 8y <= 20

   Two ways to enter the variable x in the COLUMNS section are:

     (Field:  2    3           4            5         6  )
   1.         x  const1       2.0         const2     5.0

   2.         x  const1       2.0
              x  const2       5.0

G. A mixed integer program requires the specification of which variables
   are required to be integer.  Markers are used to indicate the start
   and end of a group of integer variables.  The start marker has its
   name in field 2, 'MARKER' in field 3, and 'INTORG' in field 5.  The
   end marker has its name in field 2, 'MARKER' in field 3, and 'INTEND'
   in field 5.  These markers are placed in the COLUMNS section.

H. A specially ordered set of degree N is a collection of variables where
   at most N variables may be non-zero.  The non-zero variables must be
   contiguous (neighbours) sorted by the ascending value of their respective
   unique weights.  In lp_solve, specially ordered sets may be of any
   cardinal type 1, 2, and higher, and may be overlapping.  The number of
   variables in the set must be equal to, or exceed the cardinal SOS order.

   Below is a representation of a SOS in an MPS file, where each SOS is
   defined in its own SOS section, which should follow the BOUNDS section.

   0        1         2         3         4
    Sx CaseName  SOSName.  SOSpriority.
       CaseName  VarName1  VarWeight1..
       CaseName  VarName2  VarWeight2..

       CaseName  VarNameN  VarWeightN..

   x at the second line, position 3, defines is the order of the SOS.
   Due to limitations in the MPS format, N is restricted to the 1..9 range.
   Each SOS should be given a unique name, SOSName. lp_solve does not
   currently use case names for SOS'es and the CaseName could be any non-empty
   value.  The SOSpriority value determines the order in which multiple SOS'es
   are analysed in lp_solve.
   See also Interpolation with GAMS.

NAME SOS2test ROWS N obj L c1 L c2 E c3 COLUMNS x1 obj -1 c1 -1 x1 c2 1 x2 obj -2 c1 1 x2 c2 -3 c3 1 x3 obj -3 c1 1 x3 c2 1 x4 obj -1 c1 10 x4 c3 -3.5 x5 obj 0 RHS rhs c1 30 c2 30 BOUNDS UP BOUND x1 40 LI BOUND x4 2 UI BOUND x4 3 SOS S2 SET SOS2 10 SET x1 10000 SET x2 20000 SET x4 40000 SET x5 50000 ENDATA