Linear Functions and Constraints

Linear Functions and Constraints

This module implements linear functions (see LinearFunction) in formal variables and chained (in)equalities between them (see LinearConstraint). By convention, these are always written as either equalities or less-or-equal. For example:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
doctest:...: DeprecationWarning: The default value of 'nonnegative' will change, to False instead of True. You should add the explicit 'nonnegative=True'.
See http://trac.sagemath.org/15521 for details.
sage: f = 1 + x[1] + 2*x[2];  f     #  a linear function
1 + x_0 + 2*x_1
sage: type(f)
<type 'sage.numerical.linear_functions.LinearFunction'>

sage: c = (0 <= f);  c    # a constraint
0 <= 1 + x_0 + 2*x_1
sage: type(c)
<type 'sage.numerical.linear_functions.LinearConstraint'>

Note that you can use this module without any reference to linear programming, it only implements linear functions over a base ring and constraints. However, for ease of demonstration we will always construct them out of linear programs (see mip).

Constraints can be equations or (non-strict) inequalities. They can be chained:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: x[0] == x[1] == x[2] == x[3]
x_0 == x_1 == x_2 == x_3

sage: ieq_01234 = x[0] <= x[1] <= x[2] <= x[3] <= x[4]
sage: ieq_01234
x_0 <= x_1 <= x_2 <= x_3 <= x_4

If necessary, the direction of inequality is flipped to always write inqualities as less or equal:

sage: x[5] >= ieq_01234
x_0 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5

sage: (x[5]<=x[6]) >= ieq_01234
x_0 <= x_1 <= x_2 <= x_3 <= x_4 <= x_5 <= x_6
sage: (x[5]<=x[6]) <= ieq_01234
x_5 <= x_6 <= x_0 <= x_1 <= x_2 <= x_3 <= x_4

TESTS:

See trac ticket #12091

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: b[0] <= b[1] <= 2
x_0 <= x_1 <= 2
sage: list(b[0] <= b[1] <= 2)
[x_0, x_1, 2]
sage: 1 >= b[1] >= 2*b[0]
2*x_0 <= x_1 <= 1
sage: b[2] >= b[1] >= 2*b[0]
2*x_0 <= x_1 <= x_2
class sage.numerical.linear_functions.LinearConstraint

Bases: sage.structure.element.Element

A class to represent formal Linear Constraints.

A Linear Constraint being an inequality between two linear functions, this class lets the user write LinearFunction1 <= LinearFunction2 to define the corresponding constraint, which can potentially involve several layers of such inequalities ((A <= B <= C), or even equalities like A == B.

Trivial constraints (meaning that they have only one term and no relation) are also allowed. They are required for the coercion system to work.

Warning

This class has no reason to be instanciated by the user, and is meant to be used by instances of MixedIntegerLinearProgram.

INPUT:

  • parent – the parent, a LinearConstraintsParent_class
  • terms – a list/tuple/iterable of two or more linear functions (or things that can be converted into linear functions).
  • equality – boolean (default: False). Whether the terms are the entries of a chained less-or-equal (<=) inequality or a chained equality.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: b[2]+2*b[3] <= b[8]-5
x_0 + 2*x_1 <= -5 + x_2
equals(left, right)

Compare left and right.

OUTPUT:

Boolean. Whether all terms of left and right are equal. Note that this is stronger than mathematical equivalence of the relations.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: (x[1] + 1 >= 2).equals(3/3 + 1*x[1] + 0*x[2] >= 8/4)
True
sage: (x[1] + 1 >= 2).equals(x[1] + 1-1 >= 1-1)
False
equations()

Iterate over the unchained(!) equations

OUTPUT:

An iterator over pairs (lhs, rhs) such that the individual equations are lhs == rhs.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: eqns = 1 == b[0] == b[2] == 3 == b[3];  eqns
1 == x_0 == x_1 == 3 == x_2

sage: for lhs, rhs in eqns.equations():
...       print str(lhs) + ' == ' + str(rhs)
1 == x_0
x_0 == x_1
x_1 == 3
3 == x_2
inequalities()

Iterate over the unchained(!) inequalities

OUTPUT:

An iterator over pairs (lhs, rhs) such that the individual equations are lhs <= rhs.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: ieq = 1 <= b[0] <= b[2] <= 3 <= b[3]; ieq
1 <= x_0 <= x_1 <= 3 <= x_2

sage: for lhs, rhs in ieq.inequalities():
...       print str(lhs) + ' <= ' + str(rhs)
1 <= x_0
x_0 <= x_1
x_1 <= 3
3 <= x_2
is_equation()

Whether the constraint is a chained equation

OUTPUT:

Boolean.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: (b[0] == b[1]).is_equation()
True
sage: (b[0] <= b[1]).is_equation()
False
is_less_or_equal()

Whether the constraint is a chained less-or_equal inequality

OUTPUT:

Boolean.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: b = p.new_variable()
sage: (b[0] == b[1]).is_less_or_equal()
False
sage: (b[0] <= b[1]).is_less_or_equal()
True
is_trivial()

Test whether the constraint is trivial.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: LC = p.linear_constraints_parent()
sage: ieq = LC(1,2);  ieq
1 <= 2
sage: ieq.is_trivial()
False

sage: ieq = LC(1);  ieq
trivial constraint starting with 1
sage: ieq.is_trivial()
True
sage.numerical.linear_functions.LinearConstraintsParent(linear_functions_parent)

Return the parent for linear functions over base_ring.

The output is cached, so only a single parent is ever constructed for a given base ring.

INPUT:

OUTPUT:

The parent of the linear constraints with the given linear functions.

EXAMPLES:

sage: from sage.numerical.linear_functions import         ...       LinearFunctionsParent, LinearConstraintsParent
sage: LF = LinearFunctionsParent(QQ)
sage: LinearConstraintsParent(LF)
Linear constraints over Rational Field
class sage.numerical.linear_functions.LinearConstraintsParent_class

Bases: sage.structure.parent.Parent

Parent for LinearConstraint

Warning

This class has no reason to be instanciated by the user, and is meant to be used by instances of MixedIntegerLinearProgram. Also, use the LinearConstraintsParent() factory function.

INPUT/OUTPUT:

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: LC = p.linear_constraints_parent();  LC
Linear constraints over Real Double Field
sage: from sage.numerical.linear_functions import LinearConstraintsParent
sage: LinearConstraintsParent(p.linear_functions_parent()) is LC
True
linear_functions_parent()

Return the parent for the linear functions

EXAMPLES:

sage: LC = MixedIntegerLinearProgram().linear_constraints_parent()
sage: LC.linear_functions_parent()
Linear functions over Real Double Field
class sage.numerical.linear_functions.LinearFunction

Bases: sage.structure.element.ModuleElement

An elementary algebra to represent symbolic linear functions.

Warning

You should never instantiate LinearFunction manually. Use the element constructor in the parent instead. For convenience, you can also call the MixedIntegerLinearProgram instance directly.

EXAMPLES:

For example, do this:

sage: p = MixedIntegerLinearProgram()
sage: p({0 : 1, 3 : -8})
x_0 - 8*x_3

or this:

sage: parent = p.linear_functions_parent()
sage: parent({0 : 1, 3 : -8})
x_0 - 8*x_3

instead of this:

sage: from sage.numerical.linear_functions import LinearFunction
sage: LinearFunction(p.linear_functions_parent(), {0 : 1, 3 : -8})
x_0 - 8*x_3
dict()

Returns the dictionary corresponding to the Linear Function.

OUTPUT:

The linear function is represented as a dictionary. The value are the coefficient of the variable represented by the keys ( which are integers ). The key -1 corresponds to the constant term.

EXAMPLE:

sage: p = MixedIntegerLinearProgram()
sage: lf = p({0 : 1, 3 : -8})
sage: lf.dict()
{0: 1.0, 3: -8.0}
equals(left, right)

Logically compare left and right.

OUTPUT:

Boolean.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: (x[1] + 1).equals(3/3 + 1*x[1] + 0*x[2])
True
is_zero()

Test whether self is zero.

OUTPUT:

Boolean.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: (x[1] - x[1] + 0*x[2]).is_zero()
True
iteritems()

Iterate over the index, coefficient pairs

OUTPUT:

An iterator over the (key, coefficient) pairs. The keys are integers indexing the variables. The key -1 corresponds to the constant term.

EXAMPLES:

sage: p = MixedIntegerLinearProgram(solver = 'ppl')
sage: x = p.new_variable()
sage: f = 0.5 + 3/2*x[1] + 0.6*x[3]
sage: for id, coeff in f.iteritems():
...      print 'id =', id, '  coeff =', coeff
id = 0   coeff = 3/2
id = 1   coeff = 3/5
id = -1   coeff = 1/2
sage.numerical.linear_functions.LinearFunctionsParent(base_ring)

Return the parent for linear functions over base_ring.

The output is cached, so only a single parent is ever constructed for a given base ring.

INPUT:

  • base_ring – a ring. The coefficient ring for the linear funcitons.

OUTPUT:

The parent of the linear functions over base_ring.

EXAMPLES:

sage: from sage.numerical.linear_functions import LinearFunctionsParent
sage: LinearFunctionsParent(QQ)
Linear functions over Rational Field
class sage.numerical.linear_functions.LinearFunctionsParent_class

Bases: sage.structure.parent.Parent

The parent for all linear functions over a fixed base ring.

Warning

You should use LinearFunctionsParent() to construct instances of this class.

INPUT/OUTPUT:

See LinearFunctionsParent()

EXAMPLES:

sage: from sage.numerical.linear_functions import LinearFunctionsParent_class
sage: LinearFunctionsParent_class
<type 'sage.numerical.linear_functions.LinearFunctionsParent_class'>
set_multiplication_symbol(symbol='*')

Set the multiplication symbol when pretty-printing linear functions.

INPUT:

  • symbol – string, default: '*'. The multiplication symbol to be used.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: f = -1-2*x[0]-3*x[1]
sage: LF = f.parent()
sage: LF._get_multiplication_symbol()
'*'
sage: f
-1 - 2*x_0 - 3*x_1
sage: LF.set_multiplication_symbol(' ')
sage: f
-1 - 2 x_0 - 3 x_1
sage: LF.set_multiplication_symbol()
sage: f
-1 - 2*x_0 - 3*x_1
sage.numerical.linear_functions.is_LinearConstraint(x)

Test whether x is a linear constraint

INPUT:

  • x – anything.

OUTPUT:

Boolean.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: ieq = (x[0] <= x[1])
sage: from sage.numerical.linear_functions import is_LinearConstraint
sage: is_LinearConstraint(ieq)
True
sage: is_LinearConstraint('a string')
False
sage.numerical.linear_functions.is_LinearFunction(x)

Test whether x is a linear function

INPUT:

  • x – anything.

OUTPUT:

Boolean.

EXAMPLES:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable()
sage: from sage.numerical.linear_functions import is_LinearFunction
sage: is_LinearFunction(x[0] - 2*x[2])
True
sage: is_LinearFunction('a string')
False

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