gmpl.texi

著名的大规模线性规划求解器源码GLPK.C语言版本,可以修剪.内有详细帮助文档.

@dfn{Symbolic name} consists of alphabetic and numeric characters, the
first of which must be alphabetic. All symbolic names are distinct (case
sensitive).

@strong{Examples}

@example
alpha123
This_is_a_name
_P123_abc_321
@end example

Symbolic names are used to identify model objects (sets, parameters,
variables, constraints, objectives) and dummy indices.

All symbolic names (except names of dummy indices) must be unique,
i.e. the model description must have no objects with the same name.
Symbolic names of dummy indices must be unique within the scope, where
they are valid.

@node Numeric literals
@section Numeric literals

@dfn{Numeric literal} has the form @i{xx}@code{E}@i{syy}, where @i{xx}
is a real number with optional decimal point, @i{s} is the sign @code{+}
or @code{-}, @i{yy} is an integer decimal exponent. The letter @code{E}
is case insensitive and can be coded as @code{e}.

@strong{Examples}

@example
123
3.14159
56.E+5
.78
123.456e-7
@end example

Numeric literals are used to represent numeric quantities. They have
obvious fixed meaning.

@node String literals
@section String literals

@dfn{String literal} is a sequence of arbitrary characters enclosed
either in single quotes or in double quotes. Both these forms are
equivalent.

If the single quote is a part of a string literal enclosed in single
quotes, it must be coded twice. Analogously, if the double quote is
a part of string literal enclosed in double quotes, it must be coded
twice.

@strong{Examples}

@example
'This is a string'
"This is another string"
'1 + 2 = 3'
'That''s all'
"She said: ""No"""
@end example

String literals are used to represent symbolic quantities.

@node Keywords
@section Keywords

@dfn{Keyword} is a sequence of alphabetic characters and possibly some
special characters. All keywords fall into two categories: reserved
keywords, which cannot be used as symbolic names, and non-reserved
keywords, which being recognized by context can be used as symbolic
names.

Reserved keywords are the following:

@example
and      else     mod      union
by       if       not      within
cross    in       or
diff     inter    symdiff
div      less     then
@end example

Non-reserved keywords are described in following sections.

All the keywords have fixed meaning, which will be explained on
discussion of corresponding syntactic constructions, where the keywords
are used.

@node Delimiters
@section Delimiters

@dfn{Delimiter} is either a single special character or a sequence of
two special characters as follows:

@example
+    ^    ==   !    :    )
-    &    >=   &&   ;    [
*    <    >    ||   :=   |
/    <=   <>   .    ..   @{
**   =    !=   ,    (    @}
@end example

If delimiter consists of two characters, there must be no spaces between
the characters.

All the delimiters have fixed meaning, which will be explained on
discussion corresponding syntactic constructions, where the delimiters
are used.

@node Comments
@section Comments

For documenting purposes the model description can be provided with
@dfn{comments}, which have two different forms. The first form is
a single-line comment, which begins with the character @code{#} and
extends until end of line. The second form is a comment sequence,
which is a sequence of any characters enclosed between @code{/*} and
@code{*/}.

@strong{Examples}

@example
set s@{1..10@}; # This is a comment
/* This is another comment */
@end example

Comments are ignored by the model translator and can appear anywhere in
the model description, where white-space characters are allowed.

@node Expressions
@chapter Expressions

@menu
* Numeric expressions::
* Symbolic expressions::
* Indexing expressions and dummy indices::
* Set expressions::
* Logical expressions::
* Linear expressions::
@end menu

@dfn{Expression} is a rule for computing a value. In model description
expressions are used as constituents of certain statements.

In general case expressions consist of operands and operators.

Depending on the type of the resultant value all expressions fall into
the following categories:

@itemize @bullet
@item numeric expressions;
@item symbolic expressions;
@item indexing expressions;
@item set expressions;
@item logical expressions;
@item linear expressions.
@end itemize

@node Numeric expressions
@section Numeric expressions

@dfn{Numeric expression} is a rule for computing a single numeric value
represented in the form of floating-point number.

The primary numeric expression may be a numeric literal, dummy index,
unsubscripted parameter, subscripted parameter, built-in function
reference, iterated numeric expression, conditional numeric expression,
or another numeric expression enclosed in parentheses.

@strong{Examples}

@quotation
@multitable @columnfractions .60 .40
@item @verb{|1.23|} @tab (numeric literal)
@item @verb{|j|} @tab (dummy index)
@item @verb{|time|} @tab (unsubscripted parameter)
@item @verb{|a['May 2003',j+1]|} @tab (subscripted parameter)
@item @verb{|abs(b[i,j])|} @tab (function reference)
@item @verb{|sum{i in S diff T} alpha[i] * b[i,j]|} @tab (iterated
expression)
@item @verb{|if i in I then 2 * p else q[i+1]|} @tab (conditional
expression)
@item @verb{|(b[i,j] + .5 * c)|} @tab (parenthesized expression)
@end multitable
@end quotation

More general numeric expressions containing two or more primary numeric
expressions may be constructed by using certain arithmetic operators.

@strong{Examples}

@example
j+1
2 * a[i-1,j+1] - b[i,j]
sum@{j in J@} a[i,j] * x[j] + sum@{k in K@} b[i,k] * x[k]
(if i in I then 2 * p else q[i+1]) / (a[i,j] + 1.5)
@end example

@subheading Numeric literals

If the primary numeric expression is a numeric literal, the resultant
value is obvious.

@subheading Dummy indices

If the primary numeric expression is a dummy index, the resultant value
is current value assigned to the dummy index.

@subheading Unsubscripted parameters

If the primary numeric expression is an unsubscripted parameter (which
must be 0-dimen@-sional), the resultant value is the value of the
parameter.

@subheading Subscripted parameters

The primary numeric expression, which refers to a subscripted parameter,
has the following syntactic form:

@iftex
@quotation
@math{name[i_1,i_2,@dots,i_n],}
@end quotation

@noindent
where @math{name} is the symbolic name of the parameter, @math{i_1},
@math{i_2}, @dots, @math{i_n} are subscripts.
@end iftex

@ifnottex
@quotation
@i{name}[@i{i}1, @i{i}2, @dots{}, @i{in}],
@end quotation

@noindent
where @i{name} is the symbolic name of the parameter, @i{i}1, @i{i}2,
@dots{}, @i{in} are subscripts.
@end ifnottex

Each subscript must be a numeric or symbolic expression. The number of
subscripts in the subscript list must be the same as the dimension of
the parameter with which the subscript list is associated.

Actual values of subscript expressions are used to identify a particular
member of the parameter that determines the resultant value of the
primary expression.

@subheading Function references

In MathProg there are the following built-in functions which may be used
in numeric expressions:

@quotation
@multitable @columnfractions .25 .75
@item @verb{|abs|}(@i{x}) @tab absolute value
@item @verb{|atan|}(@i{x}) @tab trigonometric arctangent
arctan@tie{}@i{x} (in radians)
@item @verb{|atan|}(@i{y},@tie{}@i{x}) @tab trigonometric arctangent
arctan@tie{}@i{y}/@i{x} (in radians)
@item @verb{|card|}(@i{x}) @tab cardinality (the number of elements)
of set @i{x}
@item @verb{|ceil|}(@i{x}) @tab smallest integer not less than @i{x}
(``ceiling of @i{x}'')
@item @verb{|cos|}(@i{x}) @tab trigonometric cosine cos@tie{}@i{x}
(in radians)
@item @verb{|exp|}(@i{x}) @tab base-@i{e} exponential
@iftex
@math{e^x}
@end iftex
@ifnottex
@i{e}^@i{x}
@end ifnottex
@item @verb{|floor|}(@i{x}) @tab largest integer not greater than @i{x}
(``floor of @i{x}'')
@item @verb{|gmtime|}() @tab the number of seconds elapsed since
00:00:00 Jan 1, 1970,
@item @tab Coordinated Universal Time@footnote{For details
@xref{Obtaining current calendar time}.}
@item @verb{|length|}(@i{x}) @tab length of character string @i{x}
@item @verb{|log|}(@i{x}) @tab natural logarithm log@tie{}@i{x}
@item @verb{|log10|}(@i{x}) @tab common (decimal) logarithm
@iftex
@math{@log_{10}x}
@end iftex
@ifnottex
log10@tie{}@i{x}
@end ifnottex
@iftex
@item @verb{|max|}@math{(x_1,x_2,@dots,x_n)} @tab the largest of values
@math{x_1}, @math{x_2}, @dots, @math{x_n}
@item @verb{|min|}@math{(x_1,x_2,@dots,x_n)} @tab the smallest of values
@math{x_1}, @math{x_2}, @dots, @math{x_n}
@end iftex
@ifnottex
@item @verb{|max|}(@i{x}1, @dots{}, @i{xn}) @tab the largest of values
@i{x}1, @dots{}, @i{xn}
@item @verb{|min|}(@i{x}1, @dots{}, @i{xn}) @tab the smallest of values
@i{x}1, @dots{}, @i{xn}
@end ifnottex
@item @verb{|round|}(@i{x}) @tab rounding @i{x} to nearest integer
@item @verb{|round|}(@i{x},@tie{}@i{n}) @tab rounding @i{x} to @i{n}
fractional decimal digits
@item @verb{|sin|}(@i{x}) @tab trigonometric sine sin@tie{}@i{x} (in
radians)
@item @verb{|sqrt|}(@i{x}) @tab square root
@iftex
@math{@sqrt{x}}
@end iftex
@ifnottex
of @i{x}
@end ifnottex
@item @verb{|str2time|}(@i{s},@tie{}@i{f}) @tab converting character
string @i{s} to calendar time@footnote{For details @xref{Converting
character string to calendar time}.}
@item @verb{|trunc|}(@i{x}) @tab truncating @i{x} to nearest integer
@item @verb{|trunc|}(@i{x},@tie{}@i{n}) @tab truncating @i{x} to @i{n}
fractional decimal digits
@item @verb{|Irand224|}() @tab pseudo-random integer uniformly
distributed in @math{[0,2^{24})}
@item @verb{|Uniform01|}() @tab pseudo-random number uniformly
distributed in @math{[0,1)}
@item @verb{|Uniform|}(@i{a},@tie{}@i{b}) @tab pseudo-random number
uniformly distributed in [@i{a},@tie{}@i{b})
@item @verb{|Normal01|}@math{()} @tab Gaussian pseudo-random variate
with
@iftex
@math{@mu=0} and @math{@sigma=1}
@end iftex
@ifnottex
mu@tie{}=@tie{}0 and sigma@tie{}=@tie{}1
@end ifnottex
@item
@iftex
@verb{|Normal|}@math{(@mu,@sigma)}
@end iftex
@ifnottex
@verb{|Normal|}(mu,@tie{}sigma)
@end ifnottex
@tab Gaussian pseudo-random
variate with given
@iftex
@math{@mu} and @math{@sigma}
@end iftex
@ifnottex
mu and sigma
@end ifnottex
@end multitable
@end quotation

Arguments of all built-in functions, except @code{card}, @code{length},
and @code{str2time}, must be numeric expressions. The argument of
@code{card} must be a set expression. The argument of @code{length} and
both arguments of @code{str2time} must be symbolic expressions.

The resultant value of the numeric expression, which is a function
reference, is the result of applying the function to its argument(s).

Note that each pseudo-random generator function have a latent argument
(i.e. some internal state), which is changed whenever the function has
been applied. Thus, if the function is applied repeatedly even to
identical arguments, due to the side effect different resultant values
are always produced.

@subheading Iterated expressions

Iterated numeric expression is a primary numeric expression, which has
the following syntactic form:

@quotation
@var{iterated-operator} @var{indexing-expression} @var{integrand}
@end quotation

@noindent
where @var{iterated-operator} is the symbolic name of the iterated
operator to be performed (see below), @var{indexing expression} is an
indexing expression which introduces dummy indices and controls
iterating, @var{integrand} is a numeric expression that participates in
the operation.

In MathProg there are four iterated operators, which may be used in
numeric expressions:

@iftex
@quotation
@multitable @columnfractions .10 .15 .75
@item @verb{|sum|} @tab summation @tab
@math{@displaystyle@sum_{(i_1,@dots,i_n)@in@Delta}x(i_1,@dots,i_n)}
@item @verb{|prod|} @tab production @tab
@math{@displaystyle@prod_{(i_1,@dots,i_n)@in@Delta}x(i_1,@dots,i_n)}
@item @verb{|min|} @tab minimum @tab
@math{@displaystyle@min_{(i_1,@dots,i_n)@in@Delta}x(i_1,@dots,i_n)}
@item @verb{|max|} @tab maximum @tab
@math{@displaystyle@max_{(i_1,@dots,i_n)@in@Delta}x(i_1,@dots,i_n)}
@end multitable
@end quotation

@noindent
where @math{i_1}, @dots, @math{i_n} are dummy indices introduced in the
indexing expression, @math{@Delta} is the domain, a set of
@math{n}-tuples specified by the indexing expression which defines
particular values assigned to the dummy indices on performing the
iterated operation, @math{x(i_1,@dots,i_n)} is the integrand, a numeric
expression whose resultant value depends on the dummy indices.
@end iftex

@ifnottex
@quotation
@multitable @columnfractions .10 .15 .75
@item @verb{|sum|} @tab summation @tab
of @i{x}(@i{i}1,@tie{}@dots{},@tie{}@i{in}) for all
(@i{i}1,@tie{}@dots{},@tie{}@i{in})@tie{}in@tie{}D
@item @verb{|prod|} @tab production @tab
of @i{x}(@i{i}1,@tie{}@dots{},@tie{}@i{in}) for all
(@i{i}1,@tie{}@dots{},@tie{}@i{in})@tie{}in@tie{}D
@item @verb{|min|} @tab minimum @tab
of @i{x}(@i{i}1,@tie{}@dots{},@tie{}@i{in}) for all
(@i{i}1,@tie{}@dots{},@tie{}@i{in})@tie{}in@tie{}D
@item @verb{|max|} @tab maximum @tab
of @i{x}(@i{i}1,@tie{}@dots{},@tie{}@i{in}) for all
(@i{i}1,@tie{}@dots{},@tie{}@i{in})@tie{}in@tie{}D
@end multitable
@end quotation

@noindent
where @i{i}1, @dots{}, @i{in} are dummy indices introduced in the
indexing expression, D is the domain, a set of @i{n}-tuples specified