Constraints on Linear Functions Tensored with a Free Module#

Here is an example of a vector-valued linear function:

Sage

sage: mip.<x> = MixedIntegerLinearProgram('ppl')   # base ring is QQ
sage: x[0] * vector([3,4]) + 1     # vector linear function
(1, 1) + (3, 4)*x_0

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram('ppl', names=('x',)); (x,) = mip._first_ngens(1)# base ring is QQ
>>> x[Integer(0)] * vector([Integer(3),Integer(4)]) + Integer(1)     # vector linear function
(1, 1) + (3, 4)*x_0

Just like linear_functions, (in)equalities become symbolic inequalities:

Sage

sage: 3 + x[0] + 2*x[1] <= 10
3 + x_0 + 2*x_1 <= 10
sage: x[0] * vector([3,4]) + 1 <= 10
(1, 1) + (3, 4)*x_0 <= (10, 10)
sage: x[0] * matrix([[0,0,1],[0,1,0],[1,0,0]]) + x[1] * identity_matrix(3) >= 0
[0 0 0]    [x_1 0         x_0]
[0 0 0] <= [0   x_0 + x_1 0  ]
[0 0 0]    [x_0 0         x_1]

Python

>>> from sage.all import *
>>> Integer(3) + x[Integer(0)] + Integer(2)*x[Integer(1)] <= Integer(10)
3 + x_0 + 2*x_1 <= 10
>>> x[Integer(0)] * vector([Integer(3),Integer(4)]) + Integer(1) <= Integer(10)
(1, 1) + (3, 4)*x_0 <= (10, 10)
>>> x[Integer(0)] * matrix([[Integer(0),Integer(0),Integer(1)],[Integer(0),Integer(1),Integer(0)],[Integer(1),Integer(0),Integer(0)]]) + x[Integer(1)] * identity_matrix(Integer(3)) >= Integer(0)
[0 0 0]    [x_1 0         x_0]
[0 0 0] <= [0   x_0 + x_1 0  ]
[0 0 0]    [x_0 0         x_1]

class sage.numerical.linear_tensor_constraints.LinearTensorConstraint(parent, lhs, rhs, equality)[source]#

Bases: Element

Formal constraint involving two module-valued linear functions.

Note

In the code, we use “linear tensor” as abbreviation for the tensor product (over the common base ring) of a linear function and a free module like a vector/matrix space.

Warning

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

INPUT:

parent – the parent, a LinearTensorConstraintsParent_class
lhs, rhs – two sage.numerical.linear_tensor_element.LinearTensor. The left and right hand side of the constraint (in)equality.
equality – boolean (default: False). Whether the constraint is an equality. If False, it is a <= inequality.

EXAMPLES:

Sage

sage: mip.<b> = MixedIntegerLinearProgram()
sage: (b[2]+2*b[3]) * vector([1,2]) <= b[8] * vector([2,3]) - 5
(1.0, 2.0)*x_0 + (2.0, 4.0)*x_1 <= (-5.0, -5.0) + (2.0, 3.0)*x_2

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('b',)); (b,) = mip._first_ngens(1)
>>> (b[Integer(2)]+Integer(2)*b[Integer(3)]) * vector([Integer(1),Integer(2)]) <= b[Integer(8)] * vector([Integer(2),Integer(3)]) - Integer(5)
(1.0, 2.0)*x_0 + (2.0, 4.0)*x_1 <= (-5.0, -5.0) + (2.0, 3.0)*x_2

is_equation()[source]#

Whether the constraint is a chained equation

OUTPUT:

Boolean.

EXAMPLES:

Sage

sage: mip.<b> = MixedIntegerLinearProgram()
sage: (b[0] * vector([1,2]) == 0).is_equation()
True
sage: (b[0] * vector([1,2]) >= 0).is_equation()
False

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('b',)); (b,) = mip._first_ngens(1)
>>> (b[Integer(0)] * vector([Integer(1),Integer(2)]) == Integer(0)).is_equation()
True
>>> (b[Integer(0)] * vector([Integer(1),Integer(2)]) >= Integer(0)).is_equation()
False

is_less_or_equal()[source]#

Whether the constraint is a chained less-or_equal inequality

OUTPUT:

Boolean.

EXAMPLES:

Sage

sage: mip.<b> = MixedIntegerLinearProgram()
sage: (b[0] * vector([1,2]) == 0).is_less_or_equal()
False
sage: (b[0] * vector([1,2]) >= 0).is_less_or_equal()
True

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('b',)); (b,) = mip._first_ngens(1)
>>> (b[Integer(0)] * vector([Integer(1),Integer(2)]) == Integer(0)).is_less_or_equal()
False
>>> (b[Integer(0)] * vector([Integer(1),Integer(2)]) >= Integer(0)).is_less_or_equal()
True

lhs()[source]#

Return the left side of the (in)equality.

OUTPUT:

Instance of sage.numerical.linear_tensor_element.LinearTensor. A linear function valued in a free module.

EXAMPLES:

Sage

sage: mip.<x> = MixedIntegerLinearProgram()
sage: (x[0] * vector([1,2]) == 0).lhs()
(1.0, 2.0)*x_0

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('x',)); (x,) = mip._first_ngens(1)
>>> (x[Integer(0)] * vector([Integer(1),Integer(2)]) == Integer(0)).lhs()
(1.0, 2.0)*x_0

rhs()[source]#

Return the right side of the (in)equality.

OUTPUT:

Instance of sage.numerical.linear_tensor_element.LinearTensor. A linear function valued in a free module.

EXAMPLES:

Sage

sage: mip.<x> = MixedIntegerLinearProgram()
sage: (x[0] * vector([1,2]) == 0).rhs()
(0.0, 0.0)

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('x',)); (x,) = mip._first_ngens(1)
>>> (x[Integer(0)] * vector([Integer(1),Integer(2)]) == Integer(0)).rhs()
(0.0, 0.0)

sage.numerical.linear_tensor_constraints.LinearTensorConstraintsParent()[source]#

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:

linear_functions_parent – a LinearFunctionsParent_class. The type of linear functions that the constraints are made out of.

OUTPUT:

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

EXAMPLES:

Sage

sage: from sage.numerical.linear_functions import LinearFunctionsParent
sage: from sage.numerical.linear_tensor import LinearTensorParent
sage: from sage.numerical.linear_tensor_constraints import         ....:     LinearTensorConstraintsParent, LinearTensorConstraintsParent
sage: LF = LinearFunctionsParent(QQ)
sage: LT = LinearTensorParent(QQ^2, LF)
sage: LinearTensorConstraintsParent(LT)
Linear constraints in the tensor product of Vector space of dimension 2
over Rational Field and Linear functions over Rational Field

Python

>>> from sage.all import *
>>> from sage.numerical.linear_functions import LinearFunctionsParent
>>> from sage.numerical.linear_tensor import LinearTensorParent
>>> from sage.numerical.linear_tensor_constraints import         Ellipsis.:     LinearTensorConstraintsParent, LinearTensorConstraintsParent
>>> LF = LinearFunctionsParent(QQ)
>>> LT = LinearTensorParent(QQ**Integer(2), LF)
>>> LinearTensorConstraintsParent(LT)
Linear constraints in the tensor product of Vector space of dimension 2
over Rational Field and Linear functions over Rational Field

class sage.numerical.linear_tensor_constraints.LinearTensorConstraintsParent_class(linear_tensor_parent)[source]#

Bases: Parent

Parent for LinearTensorConstraint

Warning

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

INPUT/OUTPUT:

See LinearTensorConstraintsParent()

EXAMPLES:

Sage

sage: p = MixedIntegerLinearProgram()
sage: LT = p.linear_functions_parent().tensor(RDF^2);  LT
Tensor product of Vector space of dimension 2 over Real Double
Field and Linear functions over Real Double Field
sage: from sage.numerical.linear_tensor_constraints import LinearTensorConstraintsParent
sage: LTC = LinearTensorConstraintsParent(LT);  LTC
Linear constraints in the tensor product of Vector space of
dimension 2 over Real Double Field and Linear functions over
Real Double Field
sage: type(LTC)
<class 'sage.numerical.linear_tensor_constraints.LinearTensorConstraintsParent_class'>

Python

>>> from sage.all import *
>>> p = MixedIntegerLinearProgram()
>>> LT = p.linear_functions_parent().tensor(RDF**Integer(2));  LT
Tensor product of Vector space of dimension 2 over Real Double
Field and Linear functions over Real Double Field
>>> from sage.numerical.linear_tensor_constraints import LinearTensorConstraintsParent
>>> LTC = LinearTensorConstraintsParent(LT);  LTC
Linear constraints in the tensor product of Vector space of
dimension 2 over Real Double Field and Linear functions over
Real Double Field
>>> type(LTC)
<class 'sage.numerical.linear_tensor_constraints.LinearTensorConstraintsParent_class'>

Element[source]#: alias of LinearTensorConstraint

linear_functions()[source]#

Return the parent for the linear functions

OUTPUT:

Instance of sage.numerical.linear_functions.LinearFunctionsParent_class.

EXAMPLES:

Sage

sage: mip.<x> = MixedIntegerLinearProgram()
sage: ieq = (x[0] * vector([1,2]) >= 0)
sage: ieq.parent().linear_functions()
Linear functions over Real Double Field

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('x',)); (x,) = mip._first_ngens(1)
>>> ieq = (x[Integer(0)] * vector([Integer(1),Integer(2)]) >= Integer(0))
>>> ieq.parent().linear_functions()
Linear functions over Real Double Field

linear_tensors()[source]#

Return the parent for the linear functions

OUTPUT:

Instance of sage.numerical.linear_tensor.LinearTensorParent_class.

EXAMPLES:

Sage

sage: mip.<x> = MixedIntegerLinearProgram()
sage: ieq = (x[0] * vector([1,2]) >= 0)
sage: ieq.parent().linear_tensors()
Tensor product of Vector space of dimension 2 over Real Double
Field and Linear functions over Real Double Field

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('x',)); (x,) = mip._first_ngens(1)
>>> ieq = (x[Integer(0)] * vector([Integer(1),Integer(2)]) >= Integer(0))
>>> ieq.parent().linear_tensors()
Tensor product of Vector space of dimension 2 over Real Double
Field and Linear functions over Real Double Field

sage.numerical.linear_tensor_constraints.is_LinearTensorConstraint(x)[source]#

Test whether x is a constraint on module-valued linear functions.

INPUT:

x – anything.

OUTPUT:

Boolean.

EXAMPLES:

Sage

sage: mip.<x> = MixedIntegerLinearProgram()
sage: vector_ieq = (x[0] * vector([1,2]) <= x[1] * vector([2,3]))
sage: from sage.numerical.linear_tensor_constraints import is_LinearTensorConstraint
sage: is_LinearTensorConstraint(vector_ieq)
doctest:warning...
DeprecationWarning: The function is_LinearTensorConstraint is deprecated;
use 'isinstance(..., LinearTensorConstraint)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
sage: is_LinearTensorConstraint('a string')
False

Python

>>> from sage.all import *
>>> mip = MixedIntegerLinearProgram(names=('x',)); (x,) = mip._first_ngens(1)
>>> vector_ieq = (x[Integer(0)] * vector([Integer(1),Integer(2)]) <= x[Integer(1)] * vector([Integer(2),Integer(3)]))
>>> from sage.numerical.linear_tensor_constraints import is_LinearTensorConstraint
>>> is_LinearTensorConstraint(vector_ieq)
doctest:warning...
DeprecationWarning: The function is_LinearTensorConstraint is deprecated;
use 'isinstance(..., LinearTensorConstraint)' instead.
See https://github.com/sagemath/sage/issues/38184 for details.
True
>>> is_LinearTensorConstraint('a string')
False