2 Quadratic form optimization: x.dot(Q @ x) -> QuadraticForm

1 core.vectors.VectorVariable

core.vectors.VectorVariable(name, size, lb=None, ub=None, domain='continuous')

A vector of optimization variables.

VectorVariable creates and manages a collection of scalar Variable instances, providing natural indexing, slicing, and iteration.

1.1 Parameters

Name	Type	Description	Default
name	str	Base name for the vector. Elements are named “{name}[0]”, “{name}[1]”, etc.	required
size	int	Number of elements in the vector.	required
lb	float | Sequence[float] | NDArray | None	Lower bound applied to all elements (None for unbounded).	None
ub	float | Sequence[float] | NDArray | None	Upper bound applied to all elements (None for unbounded).	None
domain	DomainType	Variable type for all elements - ‘continuous’, ‘integer’, or ‘binary’.	'continuous'

1.2 Example

x = VectorVariable(“x”, 5, lb=0) x[0] # Variable named “x[0]” with lb=0 x[1:3] # VectorVariable with elements x[1], x[2] len(x) # 5 for v in x: print(v.name) # x[0], x[1], …, x[4]

1.3 Attributes

Name	Description
T	Transpose is not supported for vectors.

1.4 Methods

Name	Description
between	Create element-wise range constraints: lb <= self <= ub.
dot	Compute dot product with another vector.
eq	Element-wise == constraint.
from_numpy	Create a VectorVariable with size inferred from a NumPy array.
get_variables	Return all variables in this vector.
norm	Compute the norm of this vector.
sum	Compute sum of all elements in the vector.
to_numpy	Extract solution values as a NumPy array.

1.4.1 between

core.vectors.VectorVariable.between(lb, ub)

Create element-wise range constraints: lb <= self <= ub.

1.4.1.1 Returns

Name	Type	Description
	list[Constraint]	List of constraints covering both bounds.

1.4.2 dot

core.vectors.VectorVariable.dot(other)

Compute dot product with another vector.

Special case: if other is a MatrixVectorProduct of A @ self, this returns a QuadraticForm for optimized gradient computation. That is, x.dot(A @ x) returns QuadraticForm(x, A) which has an O(1) gradient rule instead of the general O(n) DotProduct gradient.

1.4.2.1 Parameters

Name	Type	Description	Default
other	VectorVariable | VectorExpression	Vector to compute dot product with.	required

1.4.2.2 Returns

Name	Type	Description
	DotProduct | Expression	DotProduct expression (scalar), or QuadraticForm if other is A @ self.

1.4.2.3 Example

x = VectorVariable(“x”, 3) y = VectorVariable(“y”, 3) d = x.dot(y) d.evaluate({“x[0]”: 1, “x[1]”: 2, “x[2]”: 3, “y[0]”: 4, “y[1]”: 5, “y[2]”: 6}) 32.0

import numpy as np Q = np.eye(3) qf = x.dot(Q @ x) # Returns QuadraticForm, not DotProduct

2.0.1 eq

core.vectors.VectorVariable.eq(other)

Element-wise == constraint.

2.0.1.1 Example

x = VectorVariable(“x”, 3) y = VectorVariable(“y”, 3) constraints = x.eq(y) # 3 constraints: x[i] == y[i]

2.0.2 from_numpy

core.vectors.VectorVariable.from_numpy(
    name,
    array,
    lb=None,
    ub=None,
    domain='continuous',
)

Create a VectorVariable with size inferred from a NumPy array.

The array values are not stored - this is just a convenience method to create a vector matching the array’s shape.

2.0.2.1 Parameters

Name	Type	Description	Default
name	str	Base name for the vector variables.	required
array	np.ndarray	NumPy array to match size (1D array expected).	required
lb	float | None	Lower bound for all variables.	None
ub	float | None	Upper bound for all variables.	None
domain	DomainType	Variable domain type.	'continuous'

2.0.2.2 Returns

Name	Type	Description
	VectorVariable	VectorVariable with size matching the array.

2.0.2.3 Example

import numpy as np data = np.array([1.0, 2.0, 3.0, 4.0]) x = VectorVariable.from_numpy(“x”, data, lb=0) len(x) 4

2.0.3 get_variables

core.vectors.VectorVariable.get_variables()

Return all variables in this vector.

2.0.3.1 Returns

Name	Type	Description
	list[Variable]	List of Variable instances in order.

2.0.4 norm

core.vectors.VectorVariable.norm(ord=2)

Compute the norm of this vector.

2.0.4.1 Parameters

Name	Type	Description	Default
ord	int	Order of the norm. 2 for L2 (Euclidean), 1 for L1 (Manhattan).	2

2.0.4.2 Returns

Name	Type	Description
	L2Norm | L1Norm	L2Norm or L1Norm expression (scalar).

2.0.4.3 Example

x = VectorVariable(“x”, 2) n = x.norm() # L2 norm n.evaluate({“x[0]”: 3, “x[1]”: 4}) 5.0 n1 = x.norm(1) # L1 norm n1.evaluate({“x[0]”: 3, “x[1]”: -4}) 7.0

2.0.5 sum

core.vectors.VectorVariable.sum()

Compute sum of all elements in the vector.

2.0.5.1 Returns

Name	Type	Description
	VectorSum	VectorSum expression (scalar).

2.0.5.2 Example

x = VectorVariable(“x”, 3) s = x.sum() s.evaluate({“x[0]”: 1, “x[1]”: 2, “x[2]”: 3}) 6.0

2.0.6 to_numpy

core.vectors.VectorVariable.to_numpy(solution)

Extract solution values as a NumPy array.

2.0.6.1 Parameters

Name	Type	Description	Default
solution	Mapping[str, float]	Dictionary mapping variable names to values.	required

2.0.6.2 Returns

Name	Type	Description
	np.ndarray	NumPy array of solution values in order.

2.0.6.3 Example

x = VectorVariable(“x”, 3) solution = {“x[0]”: 1.0, “x[1]”: 2.0, “x[2]”: 3.0} x.to_numpy(solution) array([1., 2., 3.])