# Xpress lp files

The Xpress LP file format provides a facility for entering a problem in a natural, algebraic LP formulation from the keyboard. The problem can be modified and saved from within lpsolve. This procedure is one way to create a file in a format that lpsolve can read. An alternative technique is to create a similar file using a standard text editor and to read it into lpsolve.

The Xpress LP format is provided as an input alternative to the MPS file format. An LP format file may be easier to generate than an MPS file if your problem already exists in an algebraic format or if you have an application that generates the problem file more readily in algebraic format (such as a C application).

Note that the Xpress LP format is not the same as the lpsolve LP format. See LP file format for a description about the native lpsolve lp format. To read/write this format from lpsolve, you need an XLI (see External Language Interfaces). The XLI for the Xpress format is called xli_Xpress.

### Options

The XLI accepts several options:

#### Reading

      -objconst      Allow constants in the objective (default).
      -noobjconst    Don't allow constants in the objective.

From the documentation of the format it is not sure if Xpress allows a constant in the objective. Tests have shown that it allows a constant as first term in the objective function. Note that the lp_solve XLI allows a constant in the objective by default and it can be anywere in the objective. Use the option -noobjconst if this should not be allowed. The parser will then give an error.

#### Writing

      -objconst      Allow constants in the objective.
      -noobjconst    Don't use constants in the objective (default).

From the documentation of the format it is not sure if Xpress allows a constant in the objective. Tests have shown that it allows a constant as first term in the objective function. The lp_solve XLI writes the constant as first term when the option -objconst is active. By default or when the option -noobjconst is used, a constant in the objective is translated to a variable objconst_term with a bound equal to the constant set to it. So no error is generated when there is a constant.

### Example

    lp_solve -rxli xli_Xpress input.lp
    lp_solve -mps input.mps -wxli xli_Xpress output.lp -wxliopt "-objconst"

### Syntax Rules of LP File Format

lpsolve will accept any problem saved in an ASCII file provided that it adheres to the following syntax rules.

1. Comments and blank lines

   Text following a backslash (\) and up to the subsequent carriage return is treated as a comment. Blank lines are ignored. Blank lines and comments may be inserted anywhere in an .lp file. For example, a common comment to put in LP files is the name of the problem:

   >     \Problem name: prob01

2. File lines, white space and identifiers

   White space and carriage returns delimit variable names and keywords from other identifiers. Keywords are case insensitive. Variable names are case sensitive. Although it is not strictly necessary, for clarity of your LP files it is perhaps best to put your section keywords on their own lines starting at the first character position on the line. No line continuation character is needed when expressions are required to span multiple lines. Lines may be broken for continuation wherever you may use white space.

3. Sections

   The LP file is broken up into sections separated by section keywords. The following are a list of section keywords you can use in your LP files. A section started by a keyword is terminated with another section keyword indicating the start of the subsequent section.

   Section keywords | Synonyms | Section contents
   --- | --- | ---
   maximize or minimize | maximum max minimum min | One linear expression describing the objective function.
   subject to | subject to: such that st s.t. st. subjectto suchthat subject such | A list of constraint expressions.
   bounds | bound | A list of bounds expressions for variables.
   integers | integer ints int | A list of variable names of integer variables. Unless otherwise specified in the bounds section, the default relaxation interval of the variables is [0, 1].
   generals | general gens gen | A list of variable names of integer variables. Unless otherwise specified in the bounds section, the default relaxation interval of the variables is [0, infinite].
   binaries | binary bins bin | A list of variable names of binary variables.
   semi-continuous | semi continuous semis semi s.c. | A list of variable names of semicontinuous variables.
   semi integer | s.i. | A list of variable names of semiinteger variables.
   partial integer | p.i. | A list of variable names of partial integer variables. Not supported by lp_solve

   Variables that do not appear in any of the variable type registration sections (i.e., integers, generals, binaries, semi-continuous, semi integer, partial integer) are defined to be continuous variables by default. That is, there is no section defining variables to be continuous variables.

   With the exception of the objective function section (maximize or minimize) and the constraints section (subject to), which must appear as the first and second sections respectively, the sections may appear in any order in the file. The only mandatory section is the objective function section. Note that you can define the objective function to be a constant in which case the problem is a so-called constraint satisfaction problem. The following two examples of LP file contents express empty problems with constant objective functions and no variables or constraints.

   Empty problem 1:

   >     Minimize

   >     End

   Empty problem 2:

   >     Minimize

   >     0

   >     End

   The end of a matrix description in an LP file can be indicated with the keyword end entered on a line by itself. This can be useful for allowing the remainder of the file for storage of comments, unused matrix definition information or other data that may be of interest to be kept together with the LP file.

4. Variable names

   Variable names can use any of the alphanumeric characters (a-z, A-Z, 0-9) and any of the following symbols:

   >     !"#$%&/,.;?@_`´{}()|~'

   A variable name can not begin with a number or a period. Care should be taken using the characters E or e since these may be interpreted as exponential notation for numbers.

5. Linear expressions

   Linear expressions are used to define the objective function and constraints. Terms in a linear expression must be separated by either a + or a - indicating addition or subtraction of the following term in the expression. A term in a linear expression is either a variable name or a numerical coefficient followed by a variable name. It is not necessary to separate the coefficient and its variable with white space or a carriage return although it is advisable to do so since this can lead to confusion. For example, the string " 2e3x" in an LP file is interpreted using exponential notation as 2000 multiplied by variable x rather than 2 multiplied by variable e3x. Coefficients must precede their associated variable names. If a coefficient is omitted it is assumed to be 1.

6. Objective function

   The objective function section can be written in a similar way to the following examples using either of the keywords maximize or minimize. Note that the keywords maximize and minimize are not used for anything other than to indicate the following linear expression to be the objective function. Note the following two examples of an LP file objective definition:

   >     Maximize
   >     - 1 x1 + 2 x2 + 3x + 4y

   or

   >     Minimize
   >     - 1 x1 + 2 x2 + 3x + 4y

   Generally objective functions are defined using many terms and thus the objective function definitions are typically always broken with line continuations. No line continuation character is required and lines may be broken for continuation wherever you may use white space.

   The objective function can be named in the same way as for constraints (see later).

7. Constraints

   The section of the LP file defining the constraints is preceded by the keyword subject to. Each constraint definition must begin on a new line. A constraint may be named with an identifier followed by a colon before the constraint expression. Constraint names must follow the same rules as variable names. If no constraint name is specified for a constraint then a default name is assigned. Constraint names are trimmed of white space before being stored. The constraints are defined as a linear expression in the variables followed by an indicator of the constraint's sense and a numerical right-hand side coefficient. The constraint sense is indicated intuitively using one of the tokens: >=, <=, or =. For example, here is a named constraint:

   >     depot01: - x1 + 1.6 x2 - 1.7 x3 <= 40

   Note that tokens > and < can be used, respectively, in place of the tokens >= and <=.

   Generally, constraints are defined using many terms so the constraint definitions are typically broken with line continuations. No line continuation character is required and lines may be broken for continuation wherever you may use white space.

8. Bounds

   The list of bounds in the bounds section are preceded by the keyword bounds. Each bound definition must begin on a new line. Single or double bounds can be defined for variables. Double bounds can be defined on the same line as 10 <= x <= 15 or on separate lines in the following ways:

   >     10 <= x
   >     15 >= x

   or

   >     x >= 10
   >     x <= 15

   If no bounds are defined for a variable, default lower and upper bounds are used. An important point to note is that the default bounds are different for different types of variables. For continuous variables the interval defined by the default bounds is [0, infinity] while for variables declared in the integers and generals section (see later) the relaxation interval defined by the default bounds is [0, 1] and [0, infinity], respectively.

   If a single bound is defined for a variable the appropriate default bounds are used as the second bound. Note that negative upper bounds on variables must be declared together with an explicit definition of the lower bound for the variable. Also note that variables can not be declared in the bounds section. That is, a variable appearing in a bounds section that does not appear in a constraint in the constraint section is ignored.

   Bounds that fix a variable can be entered as simple equalities. For example, x6 = 7.8 is equivalent to 7.8 <= x6 <= 7.8. The bounds +inf (positive infinity) and -inf (negative infinity) must be entered as strings (case insensitive):

   >     +infinity, -infinity, +inf, -inf.

   Note that the keywords infinity and inf may not be used as a right-hand side coefficient of a constraint.

   A variable with a negative infinity lower bound and positive infinity upper bound may be entered as free (case insensitive). For example, x9 free in an LP file bounds section is equivalent to:

   >     - infinity <= x9 <= + infinity

   or

   >     - infinity <= x9

   In the last example here, which uses a single bound is used for x9 (which is positive infinity for continuous example variable x9).

9. Generals, Integers and binaries

   The generals, integers and binaries sections of an LP file is used to indicate the variables that must have integer values in a feasible solution. The difference between the variables registered in each of these sections is in the definition of the default bounds that the variables will have. For variables registered in the generals section the default bounds are 0 and infinity. For variables registered in the integers section the default bounds are 0 and 1. The bounds for variables registered in the binaries section are 0 and 1.

   The lines in the generals, integers and binaries sections are a list of white space or carriage return delimited variable names. Note that variables can not be declared in these sections. That is, a variable appearing in one of these sections that does not appear in a constraint in the constraint section is ignored.

10. Semi-continuous and semi-integer

    The semi-continuous and semi integer sections of an LP file relate to two similar classes of variables and so their details are documented here simultaneously.

    The semi-continuous (or semi integer) section of an LP file are used to specify variables as semi-continuous (or semi-integer) variables, that is, as variables that may take either (a) value 0 or (b) real (or integer) values from specified thresholds and up to the variables' upper bounds.

    The lines in a semi-continuous (or semi integer) section are a list of white space or carriage return delimited entries that are either (i) a variable name or (ii) a variable name-number pair. The following example shows the format of entries in the semi-continuous section.

    >     Semi-continuous
    >     x7 >= 2.3
    >     x8
    >     x9 >= 4.5

    The following example shows the format of entries in the semi integer section.

    >     Semi integer
    >     x7 >= 3
    >     x8
    >     x9 >= 5

    Note that you can not use the <= token in place of the >= token.

    The threshold of the interval within which a variable may have real (or integer) values is defined in two ways depending on whether the entry for the variable is (i) a variable name or (ii) a variable name-number pair. If the entry is just a variable name, then the variable's threshold is the variable's lower bound, defined in the bounds section (see earlier). If the entry for a variable is a variable name-number pair, then the variable's threshold is the number value in the pair.

    It is important to note that if (a) the threshold of a variable is defined by a variable namenumber pair and (b) a lower bound on the variable is defined in the bounds section, then:

    Case 1) If the lower bound is less then zero, then the lower bound is zero.

    Case 2) If the lower bound is greater than zero but less than the threshold, then the value of zero is essentially cut off the domain of the semi-continuous (or semi-integer) variable and the variable becomes a simple bounded continuous (or integer) variable.

    Case 3) If the lower bound is greater than the threshold, then the variable becomes a simple lower bounded continuous (or integer) variable.

    If no upper bound is defined in the bounds section for a semi-continuous (or semi-integer) variable, then the default upper bound that is used is the same as for continuous variables, for semi-continuous variables, and generals section variables, for semi-integer variables.

11. Partial integers

    The partial integers section of an LP file is used to specify variables as partial integer variables, that is, as variables that can only take integer values from their lower bounds up to specified thresholds and then take continuous values from the specified thresholds up to the variables' upper bounds.

    lp_solve does not support partial integers. An error will be generated by the parser if a partial integer section is found.

12. Special ordered sets

    Special ordered sets are defined as part of the constraints section of the LP file. The definition of each special ordered set looks the same as a constraint except that the sense is always = and the right hand side is either S1 or S2 (case sensitive) depending on whether the set is to be of type 1 or 2, respectively. Special ordered sets of type 1 require that, of the non-negative variables in the set, one at most may be non-zero. Special ordered sets of type 2 require that at most two variables in the set may be non-zero, and if there are two non-zeros, they must be adjacent. Adjacency is defined by the weights, which must be unique within a set given to the variables. The weights are defined as the coefficients on the variables in the set constraint. The sorted weights define the order of the special ordered set. It is perhaps best practice to keep the special order sets definitions together in the LP file to indicate (for your benefit) the start of the special ordered sets definition with the comment line \Special Ordered Sets as is done when a problem is written to an LP file. The following example shows the definition of a type 1 and type 2 special ordered set.

    >     Sos101: 1.2 x1 + 1.3 x2 + 1.4 x4 = S1
    >     Sos201: 1.2 x5 + 1.3 x6 + 1.4 x7 = S2

### Some examples

#### Example 1

    Minimize
     COST:    XONE + 4 YTWO + 9 ZTHREE + 2
    Subject To
     LIM1:    XONE + YTWO <= 5
     LIM2:    XONE + ZTHREE >= 10
     MYEQN:   - YTWO + ZTHREE  = 7
    Bounds
     0 <= XONE <= 4
    -1 <= YTWO <= 1
    End

#### Example 2

    Minimize
    obj: - 2 x3

    Subject To
    c1: x2 - x1 <= 10
    c2: x1 + x2 + x3 <= 20

    Bounds
    x1 <= 30
    2 <= x3 <= 3

    s.i.
    x3
    x1 >= 2.1

    End

#### Example 3

    Minimize
    obj: - 2 x3

    Subject To
    c1: x2 - x1 <= 10

    \SOS
    sos101: 4 x2 + 2 x3 = S2

    c2: x1 + x2 + x3 <= 20

    sos102: x1 + x2 + x3 = S1
    sos201: 1.2 x3 +1.3 x2 + 1.4 x1 = S2

    Bounds
    x1 <= 30
    2 <= x3 <= 3

    End