# MPS format

##

MPS format

MPS format is the oldest and most widely used format for storing math programming models. There are actually two variants of this format in wide use. In fixed format, the various fields must always start at fixed columns in the file. Free format is very similar, but the fields are separated by white space instead of appearing in specific columns. One important practical difference between the two formats is in name length. In fixed format, row and column names are exactly 8 characters, and spaces are part of the name. In free format, names can be arbitrarily long (although the Gurobi reader places a 255 character limit on name length), and names may not contain spaces. The Gurobi MPS reader reads both MPS types, and recognizes the format automatically.

Note that any line that begins with the `*`

character is a
comment. The contents of that line are ignored.

NAME section

The first section in an MPS format file is the `NAME`

section.
It gives the name of the model:

NAME AFIROIn fixed format, the model name starts in column 15.

ROWS section

The next section is the `ROWS`

section. It begins with the word
`ROWS`

on its own line, and continues with one line for each row
in the model. These lines indicate the constraint type (`E`

for
equality, `L`

for less-than-or-equal, or `G`

for
greater-than-or-equal), and the constraint name. In fixed format, the
type appears in column 2 and the row name starts in column 5. Here's
a simple example:

ROWS E R09 E R10 L X05 N COSTNote that an

`N`

in the type field indicates that the row is a
*free row*. The first free row is traditionally used as the objective function, while additional free rows are ignored.

COLUMNS section

The next and typically largest section of an MPS file is the
`COLUMNS`

section, which lists the columns in the model and the
non-zero coefficients associated with each. Each line in the columns
section provides a column name, followed by either zero, one, or two
non-zero coefficients from that column. Coefficients are specified
using a row name first, followed by a floating-point value. Consider
the following example:

COLUMNS X01 X48 .301 R09 -1. X01 R10 -1.06 X05 1. X02 X21 -1. R09 1. X02 COST -4.The first line indicates that column

`X01`

has a non-zero in row
`X48`

with coefficient `.301`

, and a non-zero in row
`R09`

with coefficient `-1.0`

. Note that multiple lines
associated with the same column must be contiguous in the file.
In fixed format, the column name starts in column 5, the row name for the first non-zero starts in column 15, and the value for the first non-zero starts in column 25. If a second non-zero is present, the row name starts in column 40 and the value starts in column 50.

Integrality markers

The `COLUMNS`

section can optionally include integrality
markers. The variables introduced between a pair of markers must take
integer values. The beginning of an integer section is marked by
an `INTORG`

marker:

MARK0000 'MARKER' 'INTORG'The end of the section is marked by an

`INTEND`

marker:
MARK0000 'MARKER' 'INTEND'The first field (beginning in column 5 in fixed format) is the name of the marker (which is ignored). The second field (in column 15 in fixed format) must be equal to the string

`'MARKER'`

(including
the single quotes). The third field (in column 40 in fixed format) is
`'INTORG'`

at the start and `'INTEND'`

at the end of the
integer section.
The `COLUMNS`

section can contain an arbitrary number of such
marker pairs.

RHS section

The next section of an MPS file is the `RHS`

section, which
specifies right-hand side values. Each line in this section may
contain one or two right-hand side values.

RHS B X50 310. X51 300. B X05 80. X17 80.The first line above indicates that row

`X50`

has a right-hand
side value of `310`

, and `X51`

has a right-hand side value
of `300`

. In fixed format, the variable name for the first
bound starts in column 15, and the first bound value starts in column
25. For the second bound, the variable name starts in column 40 and
the value starts in column 50. The name of the RHS is specified in
the first field (column 5 in fixed format), but this name is ignored
by the Gurobi reader. If a row is not mentioned anywhere in the
`RHS`

section, that row takes a right-hand side value of
`0`

.
BOUNDS section

The next section in an MPS file is the optional `BOUNDS`

section. By default, each variable takes a lower bound of 0 and an
infinite upper bound. Each line in this section can modify the lower
bound of a variable, the upper bound, or both. Each line indicates a
bound type (in column 2 in fixed format), a bound name (ignored), a
variable name (in column 15 in fixed format), and a bound value (in
columns 25 in fixed format). The different bound types, and the
meaning of the associate bound value, are as follows:

LO | lower bound |

UP | upper bound |

FX | variable is fixed at the specified value |

FR | free variable (no lower or upper bound) |

MI | infinite lower bound |

PL | infinite upper bound |

BV | variable is binary (equal 0 or 1) |

LI | lower bound for integer variable |

UI | upper bound for integer variable |

SC | upper bound for semi-continuous variable |

Consider the following example:

BOUNDS UP BND X50 80. LO BND X51 20. FX BND X52 30.In this

`BOUNDS`

section, variable `X50`

gets a upper
bound of `80`

(lower bound is unchanged at `0`

,
`X51`

gets a lower bound of `20`

(infinite upper bound is
unchanged), and `X52`

is fixed at `30`

.
QUADOBJ section

The next section in an MPS file is the optional `QUADOBJ`

section, which contains quadratic objective terms. Each line in this
section represents a single non-zero value in the lower triangle of
the Q matrix. The names of the two variable that participate in the
quadratic term are found first (starting in columns 5 and 15 in fixed
format), followed by the numerical value of the coefficient (in column
25 in fixed format). By convention, the Q matrix has an implicit
one-half multiplier associated with it. Here's an example containing
three quadratic terms:

QUADOBJ X01 X01 10.0 X01 X02 2.0 X02 X02 2.0These three terms would represent the quadratic function (recall that the single off-diagonal term actually represents a pair of non-zero values in the symmetric Q matrix).

QCMATRIX section

The next section in an MPS file contains zero or more `QCMATRIX`

blocks. These blocks contain the quadratic terms associated with the
quadratic constraints. There should be one block for each quadratic
constraint in the model.

Each `QCMATRIX`

block starts with a line that indicates the name
of the associated quadratic constraint (starting in column 12 in fixed
format). This is followed by one of more quadratic terms. Each term
is described on one line, which gives the names of the two
involved variables (starting in columns 5 and 15 in fixed format),
followed by the coefficient (in column 25 in fixed format).
For example:

QCMATRIX QC0 X01 X01 10.0 X01 X02 2.0 X02 X01 2.0 X02 X02 2.0These four lines describe three quadratic terms: quadratic constraint

`QC0`

contains terms , , and .
Note that a `QCMATRIX`

block must contain a symmetric matrix, so
for example an `X01*X02`

term must be accompanied by a matching
`X02*X01`

term.
Linear terms for quadratic constraint `QC0`

appear in the
`COLUMNS`

section. The sense and right-hand side value appear in
the `ROWS`

and `RHS`

sections, respectively.

PWLOBJ section

The next section in an MPS file is the optional `PWLOBJ`

section,
which contains piecewise-linear objective functions. Each line in
this section represents a single point in a piecewise-linear objective
function. The name of the associated variable appears first (starting
in column 4), followed by the x and y coordinates of the point
(starting in columns 14 and 17). Here's an example containing two
piecewise-linear expressions, for variables `X01`

and `X02`

,
each with three points:

X01 1 1 X01 2 2 X01 3 4 X02 1 1 X02 3 5 X02 7 10

SOS section

The next section in an MPS file is the optional `SOS`

section.
The representation for a single SOS constraint contains one line that
provides the type of the SOS set (`S1`

for SOS type 1 or
`S2`

for SOS type 2, found in column 2 in fixed format) and the
name of the SOS set (column 5 in fixed format) of the SOS set. This
is followed by one line for each SOS member. The member line gives
the name of the member (column 5 in fixed format) and the associated
weight (column 15 in fixed format). Here's an example containing two
SOS2 sets.

SOS S2 sos1 x1 1 x2 2 x3 3 S2 sos2 x3 1 x4 2 x5 3

ENDATA

The final line in an MPS file must be an `ENDATA`

statement.

Additional notes

Note that in the Gurobi optimizer, MPS models are always written in full precision. That means that if you write a model and then read it back, the data associated with the resulting model will be bit-for-bit identical to the original data.