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LP format

LP format

The LP format captures an optimization model in a way that is easier for humans to read than MPS format, and can often be more natural to produce. One limitation of the LP format is that it doesn't preserve several model properties. In particular, LP files do not preserve column order when read, and they typically don't preserve the exact numerical values of the coefficients (although this isn't inherent to the format).

Unlike MPS files, LP files do not rely on fixed field widths. Line breaks and white space characters are used to separate objects. Here is a simple example:

\ LP format example

  x + y + z
Subject To
  c0: x + y = 1
  c1: x + 5 y + 2 z <= 10
  qc0: x + y + [ x ^ 2 - 2 x * y + 3 y ^ 2 ] <= 5
  0 <= x <= 5
  z >= 2
  x y z

The backslash symbol starts a comment; the remainder of that line is ignored.

Variable names play a major role in LP files. Each variable must have its own unique name. A name should be no longer than 255 characters, and to avoid confusing the LP parser, it should not begin with a number, or contain any of the characters +, -, *, ^, <, >, =, (, ), [, ], ,, or :.

Note that white space is not optional in the Gurobi LP format. Thus, for example, the text x+y+z would be treated as a single variable name, while x + y + z would be treated as a three term expression.

LP files are structured as a list of sections, where each section captures a logical piece of the whole optimization model. Sections begin with particular keywords, and must generally come in a fixed order, although a few are allowed to be interchanged.

Objective Section

The first section in an LP file is the objective section. This section begins with one of the following six keywords: minimize, maximize, minimum, maximum, min, or max. Capitalization is ignored. This keyword may appear alone, or it may be immediately followed by multi-objectives, which indicates that the model contains multiple objective functions.

Single-Objective Case

Let us consider single-objective models first, where this header is followed by a single linear or quadratic expression that captures the objective function.

The objective optionally begins with a label. A label consists of a name, followed by a colon character, following by a space. A space is allowed between the name and the colon, but not required.

The objective then continues with a list of linear terms, separated by the + or - operators. A term can contain a coefficient and a variable (e.g., 4.5 x), or just a variable (e.g., x). The objective can be spread over many lines, or it may be listed on a single line. Line breaks can come between tokens, but never within tokens.

The objective may optionally continue with a list of quadratic terms. The quadratic portion of the objective expression begins with a [ symbol and ends with a ] symbol, followed by / 2. These brackets should enclose one or more quadratic terms. Either squared terms (e.g., 2 x ^ 2) or product terms (e.g., 3 x * y) are accepted. Coefficients on the quadratic terms are optional.

For variables with piecewise-linear objective functions, the objective section will include a __pwl(x) term, where x is the name of the variable. You should view these as comments; they are ignored by the LP reader. The actual piecewise-linear expressions are pulled from the later PWLObj section.

The objective expression must always end with a line break.

An objective section might look like the following:

  obj: 3.1 x + 4.5 y + 10 z + [ x ^ 2 + 2 x * y + 3 y ^ 2 ] / 2

Multi-Objective Case

In the multi-objective case, the header is followed by one or more linear objective functions, where each starts with its own sub-header. The sub-header gives the name of the objective, followed by a number of optional fields that provide a Priority, Weight, absolute tolerance (AbsTol) and relative tolerance (RelTol) for that objective (see ObjNPriority, ObjNWeight, ObjNRelTol, and ObjNAbsTol for details on the meanings of these fields). The fields start with the field name, followed by a =, followed by the value. For example:

  OBJ0: Priority=2 Weight=1 AbsTol=0 RelTol=0
Please refer to the multi-objective section for additional details.

Each sub-header is followed by a linear expression that captures that objective.

A complete multi-objective section might look like the following:

Minimize multi-objectives
  OBJ0: Priority=2 Weight=1 AbsTol=0 RelTol=0
    3.1 x + 4.5 y + 10 z
  OBJ1: Priority=1 Weight=1 AbsTol=0 RelTol=0
    10 x + 0.1 y

The objective section is optional. The objective is set to 0 when it is not present.

Constraints Section

The next section is the constraints section. It begins with one of the following headers, on its own line: subject to, such that, st, or s.t.. Capitalization is ignored.

The constraint section can have an arbitrary number of constraints. Each constraint starts with an optional label (constraint name, followed by a colon, followed by a space), continues with a linear expression, followed by an optional quadratic expression (enclosed in square brackets), and ends with a comparison operator, followed by a numerical value, followed by a line break. Valid comparison operators are =, <=, <, >=, or >. Note that LP format does not distinguish between strict and non-strict inequalities, so for example < and <= are equivalent.

Note that the left-hand side of a constraint may not contain a constant term; the constant must appear on the right-hand side.

The following is a simple example of a valid linear constraint:

  c0: 2.5 x + 2.3 y + 5.3 z <= 8.1
The following is a valid quadratic constraint:
  qc0: 3.1 x + 4.5 y + 10 z + [ x ^ 2 + 2 x * y + 3 y ^ 2 ] <= 10

The constraint section may also contain another constraint type: the so-called indicator constraint. Indicator constraints start with an optional label (constraint name, followed by a colon, followed by a space), followed by a binary variable, a space, a =, again a space and a value, either 0 or 1. They continue with a space, followed by ->, and again a space and finally a linear constraint (without a label).

For example:

  c0: b1 = 1 -> 2.5 x + 2.3 y + 5.3 z <= 8.1

This example constraint requires the given linear constraint to be satisfied if the variable b1 takes a value of 1.

Every LP format file must have a constraints section.

Lazy Constraints Section

The next section is the lazy constraints section. It begins with the line Lazy Constraints, and continues with a list of linear constraints in the exact same format as the linear constraints in the constraints section. For example:

Lazy Constraints
  c0: 2.5 x + 2.3 y + 5.3 z <= 8.1

Lazy constraints are handled differently from other constraints by the MIP solver. A lazy constraint only becomes active when the MIP solver finds a candidate solution that violates the constraint.

This section is optional.

Bounds Section

The next section is the bounds section. It begins with the word Bounds, on its own line, and is followed by a list of variable bounds. Each line specifies the lower bound, the upper bound, or both for a single variable. The keywords inf or infinity can be used in the bounds section to specify infinite bounds. A bound line can also indicate that a variable is free, meaning that it is unbounded in either direction.

Here are examples of valid bound lines:

  0 <= x0 <= 1
  x1 <= 1.2
  x2 >= 3
  x3 free
  x2 >= -Inf

It is not necessary to specify bounds for all variables; by default, each variable has a lower bound of 0 and an infinite upper bound. In fact, the entire bounds section is optional.

Variable Type Section

The next section is the variable types section. Variables can be designated as being either binary, general integer, or semi-continuous. In all cases, the designation is applied by first providing the appropriate header (on its own line), and then listing the variables that have the associated type. For example:

  x y z
Variable type designations don't need to appear in any particular order (e.g., general integers can either precede or follow binaries). If a variable is included in multiple sections, the last one determines the variable type.

Valid keywords for variable type headers are: binary, binaries, bin, general, generals, gen, semi-continuous, semis, or semi.

The variable types section is optional. By default, variables are assumed to be continuous.

SOS Section

An LP file can contain a section that captures SOS constraints of type 1 or type 2. The SOS section begins with the SOS header on its own line (capitalization isn't important). An arbitrary number of SOS constraints can follow. An SOS constraint starts with a name, followed by a colon (unlike linear constraints, the name is not optional here). Next comes the SOS type, which can be either S1 or S2. The type is followed by a pair of colons.

Next come the members of the SOS set, along with their weights. Each member is captured using the variable name, followed by a colon, followed by the associated weight. Spaces can optionally be placed before and after the colon. An SOS constraint must end with a line break.

Here's an example of an SOS section containing two SOS constraints:

  sos1: S1 :: x1 : 1  x2 : 2  x3 : 3
  sos2: S2 :: x4:8.5  x5:10.2  x6:18.3

The SOS section is optional.

PWLObj Section

An LP file can contain a section that captures piecewise-linear objective functions. The PWL section begins with the PWLObj header on its own line (capitalization isn't important). Each piecewise-linear objective function is associated with a model variable. A PWL function starts with the corresponding variable name, followed immediately by a colon (the name is not optional). Next come the points that define the piecewise-linear function. These points are represented as (x, y) pairs, with parenthesis surrounding the two values and a comma separating them. A PWL function must end with a line break.

Here's an example of a PWLObj section containing two simple piecewise-linear functions:

  x1: (1, 1) (2, 2) (3, 4)
  x2: (1, 3) (3, 5) (100, 300)

The PWLObj section is optional.

General Constraint Section

An LP file may contain a section that captures more general constraints. The general constraint section starts with one of the following keywords general constraints, general constraint, gencons, or g.c.. Capitalization is ignored.

Each general constraint starts with an optional label (constraint name, followed by a colon, followed by a space), continues with a variable name, the so-called resultant, followed by a space, a equation character =, and a space again. The line continues with a general constraint type specifier, such as MIN, MAX, OR, AND, or ABS followed by a space and a ( and a space again. Capitalization is ignored.

What follows depends on the general constraint type. MIN or MAX constraints expect a non empty list of variables or values separated by a space, a comma, and space again. OR and AND constraints expect a list of binary variables, again separated by a space, a comma, and space again. ABS constraints only expect one variable name.

All these general constraints end with a space, ), and a line break.

The other general constraint type, the INDICATOR constraint, appears in the constraints section, which is described above.

The following is an example of a general constraint section :

General Constraints
 gc0: r1 = MAX ( x1 , x2 , x10 , 0.7 )
 gencons1: r2 = MIN ( y0 , 10 , y1 , r1 )
 and1: r = AND ( b1 , b2 )
 or1: r = OR ( b3 , b4 )
 GC14: xabs = ABS ( x )

This section is optional.

For more information, consult the general constraint discussion.

End statement

The last line in an LP format file should be an End statement.