opengen.constraints package

Submodules

opengen.constraints.affine_space module

class opengen.constraints.affine_space.AffineSpace(A, b)

Bases: Constraint

An affine constraint

A constraint of the form \(Ax = b\), where \(A\) and \(b\) are a matrix and a vector of appropriate dimensions

__init__(A, b)

Constructor for an affine space

Returns:: new instance of AffineSpace

distance_squared(u)

Squared distance to affine space

Not implemented yet

is_compact(): Affine spaces are not compact sets

is_convex(): Affine spaces are convex sets

property matrix_a: Matrix A

project(u)

Projection on affine space

Not implemented yet

property vector_b: Vector b

opengen.constraints.ball1 module

class opengen.constraints.ball1.Ball1(center=None, radius: float = 1.0)

Bases: Constraint

Ball1 aka Norm-1 Ball

Ball-1 with given radius, that is

\(\mathcal{B}_{1}(x_0, r) = \{x\in{\rm I\!R}^n {}:{} \|x - x_0\|_1 \leq r\}\)

__init__(center=None, radius: float = 1.0)

Constructor for a Ball1

Parameters:: radius – ball radius (default: 1)
Returns:: New instance of Ball1 with given radius

property center: Returns the center of the ball

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project on the current Ball-1

Parameters:: u – vector u
Returns:: projection of u onto the current ball-1

property radius: Returns the radius of this ball

opengen.constraints.ball2 module

class opengen.constraints.ball2.Ball2(center=None, radius: float = 1.0)

Bases: Constraint

A Euclidean ball constraint

A constraint of the form \(\|u-u_0\| \leq r\), where \(u_0\) is the center of the ball and r is its radius

__init__(center=None, radius: float = 1.0)

Constructor for a Euclidean ball constraint

Parameters:

center – center of the ball; if this is equal to Null, the ball is centered at the origin
radius – radius of the ball

Returns:

New instance of Ball2 with given center and radius

property center: Returns the center of the ball

distance_squared(u)

Computes the squared distance between a given point u and this ball

Parameters:: u – given point; can be a list of float, a numpy n-dim array (ndarray) or a CasADi SX/MX symbol
Returns:: distance from set as a float or a CasADi symbol

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

property radius: Returns the radius of the ball

opengen.constraints.ball_inf module

class opengen.constraints.ball_inf.BallInf(center=None, radius: float = 1.0)

Bases: Constraint

Norm-ball of norm infinity translated by given vector

Centered inf-ball around given point, that is

\(\mathcal{B}_{\infty}(x_0, r) = \{x\in{\rm I\!R}^n {}:{} \|x - x_0\|_{\infty} \leq r\}\)

__init__(center=None, radius: float = 1.0)

Constructor for an infinity ball constraint

Parameters:

center – center of the ball; if this is equal to Null, the ball is centered at the origin
radius – radius of the ball

Returns:

New instance of Ballinf with given center and radius

property center: Returns the center of the ball

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

property radius: Returns the radius of the ball

opengen.constraints.cartesian module

class opengen.constraints.cartesian.CartesianProduct(segments: List[int], constraints: List[Constraint])

Bases: Constraint

__init__(segments: List[int], constraints: List[Constraint])

Cartesian product

\(X = X_1 \times X_1 \times \ldots \times X_s.\)

Construct a Cartesian product of constraints by providing a list of sets and their dimensions as follows: an n-dimensional vector x can be partitioned into subvectors as \(x = (x_{[1]}, x_{[2]}, ..., x_{[s]})\), where each \(x_{[i]}\) has dimension \(m_i\).

For example consider the 5-dimensional vector \(x = (x_0, x_1, x_2, x_3, x_4)\), which can be partitioned into \(x_{[1]} = (x_0, x_1)\) and \(x_{[2]} = (x_2, x_3, x_4)\). We can associate with \(x_{[1]}\) the indices [0, 1] and with \(x_{[2]}\) the indices [2, 3, 4]. The segment ids are the indices 1 and 4.

Example:

In this example we shall define the set \(X = \mathcal{B}_{1.5} \times R \times {\rm I\!R}^{5}\), where \(\mathcal{B}_{1.5}\) is a Euclidean ball of dimension 2 with radius 1.5, \(R\) is a 3-dimensional rectangle with \(x_{\min} = (-1, -2, -3)\) and \(x_{\max} = (0, 10, -1)\). Here the segments are [1, 4, 9].

>>> ball = og.constraints.Ball2(None, 1.5)
>>> rect = og.constraints.Rectangle(xmin=[-1,-2,-3], xmax=[0, 10, -1])
>>> free = og.constraints.NoConstraints()
>>> segment_ids = [1, 4, 9]
>>> my_set = og.constraints.CartesianProduct(segment_ids, [ball, rect, free])

Parameters:

segments – ids of segments
constraints – list of sets

property constraints

Returns:: list of constraints comprising the current instance of CartesianProduct

distance_squared(u)

Squared distance of given vector, u, from the current instance of CartesianProduct

Parameters:: u – vector u
Returns:: squared distance (float)

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

segment_dimension(i)

Dimension of segment i

Parameters:: i – index of segment (starts at 0)
Returns:: dimension of i-th index

property segments

Returns:: list of segments

opengen.constraints.constraint module

class opengen.constraints.constraint.Constraint

Bases: object

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

opengen.constraints.finite_set module

class opengen.constraints.finite_set.FiniteSet(points=None)

Bases: Constraint

Finite set

A set of the form \(A = \{a_1, a_2, \ldots, a_K\}\)

__init__(points=None)

Constructor for a finite set

Parameters:: points (list of lists) – the elements of the set

cardinality(): Cardinality of the set

dimension(): Dimension of the set

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

property points: List of points of this set

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

opengen.constraints.halfspace module

class opengen.constraints.halfspace.Halfspace(normal_vector, offset: float)

Bases: Constraint

Halfspace constraint

A halfspace is a set of the form \(H = \{c^\intercal x \leq b\}\), where c is a given vector and b is a constant scalar.

__init__(normal_vector, offset: float)

Construct a new halfspace \(H = \{c^\intercal x \leq b\}\)

Parameters:

normal_vector – vector c
offset – parameter b

dimension()

Dimension of the halfspace

Returns:: length/dimension of normal vector

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact()

Whether the set is compact

H is compact iff \(b < 0\) and \(c = 0\), in which case H is empty

is_convex()

A halfspace is a convex set

Returns:: True

property normal_vector: Vector c

property offset: Parameter b

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

opengen.constraints.no_constraints module

class opengen.constraints.no_constraints.NoConstraints

Bases: Constraint

This is the entire \({\rm I\!R}^n\)

__init__()

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

opengen.constraints.rectangle module

class opengen.constraints.rectangle.Rectangle(xmin, xmax)

Bases: Constraint

A Rectangle (Box) constraint

A set of the form

\(X = \{x\in{\mathrm{I\!R}}^n {}:{} x_{\mathrm{min}} \leq x \leq x_{\mathrm{max}}\}\)

__init__(xmin, xmax)

Construct a new instance of Rectangle

Parameters:

xmin – minimum bounds (can be None)
xmax – maximum bounds (can be None)

Raises:

Exception – if both xmin and xmax is None
Exception – if xmin and xmax are both not None and not a list
Exception – if xmin and xmax have incompatible lengths
Exception – if xmin(i) > xmax(i) for some i (empty set)

Returns:

A new instance of Rectangle

dimension(): Dimension of this rectangle (inferred by the dimensions of xmin and xmax)

distance_squared(u)

Squared distance to this rectangle

Parameters:: u – given point
Returns:: squared distance of u to this rectangle

idx_bound_finite_all()

Coordinates where both bounds are finite

Returns:: list of coordinates

idx_infinite_only_xmax()

Coordinates at which xmax is infinity and xmin is finite

Returns:: list of coordinates

idx_infinite_only_xmin()

Coordinates at which xmin is minus infinity and xmax is finite

Returns:: list of coordinates

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

is_orthant()

Whether this rectangle is an orthant

An orthant is a rectangle whose projection on every coordinate is an interval of the form \([0, \infty)\) or \((-\infty, 0]\).

Return type:: boolean

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

property xmax: Maximum bound

property xmin: Minimum bound

opengen.constraints.simplex module

class opengen.constraints.simplex.Simplex(alpha: float = 1.0)

Bases: Constraint

Simplex constraint

A set of the form

\(X = \left\{x \in \mathrm{I\!R}^n : \sum_{i=1}^{n} x_i = \alpha, x \geq 0\right\}\)

where \(\alpha\) is a positive constant.

__init__(alpha: float = 1.0)

Constructor for a Simplex constraint

Parameters:: alpha – size parameter of simplex (default: 1)
Returns:: new instance of Simplex with given alpha

property alpha: Returns the simplex value alpha

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact (True)

is_convex(): Whether the set is convex (True)

project(y)

Computes the projection of a given point \(y\in{\rm I\!R}^n\) onto the current simplex.

Parameters:: y – given point; must be a list of numbers (float, int) or a numpy n-dim array (ndarray)
Returns:: the projection point in \({\rm I\!R}^n\) as a numpy array of float64s

opengen.constraints.soc module

class opengen.constraints.soc.SecondOrderCone(a: float = 1.0)

Bases: Constraint

A Second-Order Cone given by \(C = \{u = (x, r): a\|x\| \leq r\}\)

Second-order cones are used in conic optimisation to describe inequalities that involve quadratic terms

__init__(a: float = 1.0)

Constructor for a Second-Order Cone set

Parameters:: a – parameter a
Returns:: New instance of a SOC with given parameter a

property a: Returns the value of parameter a

distance_squared(u)

Computes the squared distance between a given point u and this second-order cone

param u:

given point; can be a list of float, a numpy n-dim array (ndarray) or a CasADi SX/MX symbol

return:

distance from set as a float or a CasADi symbol

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

opengen.constraints.sphere2 module

class opengen.constraints.sphere2.Sphere2(center=None, radius: float = 1.0)

Bases: Constraint

A Euclidean sphere constraint

A constraint of the form \(\|u-u_0\| = r\), where \(u_0\) is the center of the ball and r is its radius

__init__(center=None, radius: float = 1.0)

Constructor for a Euclidean sphere constraint

Parameters:

center – center of the sphere; if this is equal to Null, the sphere is centered at the origin
radius – radius of the sphere

Returns:

New instance of Sphere2 with given center and radius

property center: Returns the center of the ball

distance_squared(u)

Computes the squared distance between a given point u and this sphere

Parameters:: u – given point; can be a list of float, a numpy n-dim array (ndarray) or a CasADi SX/MX symbol
Returns:: distance from set as a float or a CasADi symbol

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

property radius: Returns the radius of the sphere

opengen.constraints.zero module

class opengen.constraints.zero.Zero

Bases: Constraint

A set that contains only the origin

The singleton \(\{0\}\)

__init__(): Constructor for set \(Z = \{0\}\)

distance_squared(u)

Squared distance of a given point to the set

Parameters:: u – given point
Returns:: squared distance
Return type:: float

is_compact(): Whether the set is compact

is_convex(): Whether the set is convex

project(u)

Project a given point onto the set

Parameters:: u – given point
Returns:: projection of u onto this set

opengen.constraints package

Submodules

opengen.constraints.affine_space module

opengen.constraints.ball1 module

opengen.constraints.ball2 module

opengen.constraints.ball_inf module

opengen.constraints.cartesian module

opengen.constraints.constraint module

opengen.constraints.finite_set module

opengen.constraints.halfspace module

opengen.constraints.no_constraints module

opengen.constraints.rectangle module

opengen.constraints.simplex module

opengen.constraints.soc module

opengen.constraints.sphere2 module

opengen.constraints.zero module

Module contents