# 14.2.12 Class Domain¶

- mosek.fusion.Domain¶
The

`Domain`

class defines a set of static method for creating various variable and constraint domains. A`Domain`

object specifies a subset of \(\real^n\), which can be used to define the feasible domain of variables and expressions.For further details on the use of these, see

`Model.Variable`

and`Model.Constraint`

.- Static members
Domain.Axis – Set the dimension along which the cones are created.

Domain.Binary – Creates a domain of binary variables.

Domain.EqualsTo – Defines the domain consisting of a fixed point.

Domain.GreaterThan – Defines the domain specified by a lower bound in each dimension.

Domain.InDExpCone – Defines the dual exponential cone.

Domain.InDGeoMeanCone – Defines the domain of dual geometric mean cones.

Domain.InDPowerCone – Defines the dual power cone.

Domain.InPExpCone – Defines the primal exponential cone.

Domain.InPGeoMeanCone – Defines the domain of primal geometric mean cones.

Domain.InPPowerCone – Defines the primal power cone.

Domain.InPSDCone – Creates a domain of Positive Semidefinite matrices.

Domain.InQCone – Defines the domain of quadratic cones.

Domain.InRange – Creates a domain specified by a range in each dimension.

Domain.InRotatedQCone – Defines the domain of rotated quadratic cones.

Domain.InSVecPSDCone – Creates a domain of vectorized Positive Semidefinite matrices.

Domain.Integral – Creates a domain of integral variables.

Domain.IsTrilPSD – Creates a domain of Positive Semidefinite matrices.

Domain.LessThan – Defines the domain specified by an upper bound in each dimension.

Domain.Sparse – Use a sparse representation.

Domain.Symmetric – Impose symmetry on a given linear domain.

Domain.Unbounded – Creates a domain in which variables are unbounded.

- Domain.Axis¶
ConeDomain Domain.Axis(ConeDomain c, int a)

Set the dimension along which the cones are created. If this conic domain is used for a variable or expression of dimension \(d\), then the conic constraint will be applicable to all vectors obtained by fixing the coordinates other than \(a\)-th and moving along the \(a\)-th coordinate. If \(d=2\) this can be used to define the conditions “every row of the matrix is in a cone” and “every column of a matrix is in a cone”.

The default is the last dimension \(a=d-1\).

- Parameters
`c`

(`ConeDomain`

) – A conic domain.`a`

(`int`

) – The axis.

- Return

- Domain.Binary¶
RangeDomain Domain.Binary(int n) RangeDomain Domain.Binary(int m, int n) RangeDomain Domain.Binary(int[] dims) RangeDomain Domain.Binary()

Create a domain of binary variables. A binary domain can only be used when creating variables, but is not allowed in a constraint. Another way of restricting variables to be integers is the method

`Variable.MakeInteger`

.- Parameters
`n`

(`int`

) – Dimension size.`m`

(`int`

) – Dimension size.`dims`

(`int`

[]) – A list of dimension sizes.

- Return

- Domain.EqualsTo¶
LinearDomain Domain.EqualsTo(double b) LinearDomain Domain.EqualsTo(double b, int n) LinearDomain Domain.EqualsTo(double b, int m, int n) LinearDomain Domain.EqualsTo(double b, int[] dims) LinearDomain Domain.EqualsTo(double[] a1) LinearDomain Domain.EqualsTo(double[,] a2) LinearDomain Domain.EqualsTo(double[] a1, int[] dims) LinearDomain Domain.EqualsTo(Matrix mx)

Defines the domain consisting of a fixed point.

- Parameters
`b`

(`double`

) – A single value. This is scalable: it means that each element in the variable or constraint is fixed to \(b\).`n`

(`int`

) – Dimension size.`m`

(`int`

) – Dimension size.`dims`

(`int`

[]) – A list of dimension sizes.`a1`

(`double`

[]) – A one-dimensional array of bounds. The shape must match the variable or constraint with which it is used.`a2`

(`double`

[,]) – A two-dimensional array of bounds. The shape must match the variable or constraint with which it is used.`mx`

(`Matrix`

) – A matrix of bound values. The shape must match the variable or constraint with which it is used.

- Return

- Domain.GreaterThan¶
LinearDomain Domain.GreaterThan(double b) LinearDomain Domain.GreaterThan(double b, int n) LinearDomain Domain.GreaterThan(double b, int m, int n) LinearDomain Domain.GreaterThan(double b, int[] dims) LinearDomain Domain.GreaterThan(double[] a1) LinearDomain Domain.GreaterThan(double[,] a2) LinearDomain Domain.GreaterThan(double[] a1, int[] dims) LinearDomain Domain.GreaterThan(Matrix mx)

Defines the domain specified by a lower bound in each dimension.

- Parameters
`b`

(`double`

) – A single value. This is scalable: it means that each element in the variable or constraint is greater than or equal to \(b\).`n`

(`int`

) – Dimension size.`m`

(`int`

) – Dimension size.`dims`

(`int`

[]) – A list of dimension sizes.`a1`

(`double`

[]) – A one-dimensional array of bounds. The shape must match the variable or constraint with which it is used.`a2`

(`double`

[,]) – A two-dimensional array of bounds. The shape must match the variable or constraint with which it is used.`mx`

(`Matrix`

) – A matrix of bound values. The shape must match the variable or constraint with which it is used.

- Return

- Domain.InDExpCone¶
ConeDomain Domain.InDExpCone() ConeDomain Domain.InDExpCone(int m) ConeDomain Domain.InDExpCone(int[] dims)

Defines the domain of dual exponential cones:

\[\left\{ x\in \real^3 ~:~ x_1 \geq -x_3 e^{-1} e^{x_2/x_3},\ x_1> 0,\ x_3< 0 \right\}\]The conic domain scales as follows. If a variable or expression constrained to an exponential cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.InDGeoMeanCone¶
ConeDomain Domain.InDGeoMeanCone() ConeDomain Domain.InDGeoMeanCone(int n) ConeDomain Domain.InDGeoMeanCone(int m, int n) ConeDomain Domain.InDGeoMeanCone(int[] dims)

Defines the domain of dual geometric mean cones:

\[\left\{ x\in \real^n ~:~ (n-1) \left(\prod_{i=1}^{n-1} x_i\right)^{1/(n-1)} \geq |x_n|,\ x_1,\ldots,x_{n-1}\geq 0 \right\}\]The conic domain scales as follows. If a variable or expression constrained to a cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`n`

(`int`

) – The size of each cone; at least 2.`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.InDPowerCone¶
ConeDomain Domain.InDPowerCone(double alpha) ConeDomain Domain.InDPowerCone(double alpha, int m) ConeDomain Domain.InDPowerCone(double alpha, int[] dims) ConeDomain Domain.InDPowerCone(double[] alphas) ConeDomain Domain.InDPowerCone(double[] alphas, int m) ConeDomain Domain.InDPowerCone(double[] alphas, int[] dims)

Defines the domain of dual power cones. For a single double argument

`alpha`

it defines the set\[\left\{ x\in \real^n ~:~ \left(\frac{x_1}{\alpha}\right)^\alpha \left(\frac{x_2}{1-\alpha}\right)^{1-\alpha} \geq \sqrt{\sum_{i=3}^n x_i^2},\ x_1,x_2\geq 0 \right\}.\]For an array

`alphas`

of length \(n_l\), consisting of weights for the cone, it defines the set\[\left\{ x\in\real^n~:~ \prod_{i=1}^{n_l}\left(\frac{x_i}{\beta_i}\right)^{\beta_i} \geq \sqrt{x_{n_l+1}^2+\cdots+x_n^2},\ x_1,\ldots,x_{n_l}\geq 0 \right\}.\]where \(\beta_i\) are the weights normalized to add up to \(1\), ie. \(\beta_i=\alpha_i/(\sum_j \alpha_j)\) for \(i=1,\ldots,n_l\).

The conic domain scales as follows. If a variable or expression constrained to a power cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`alpha`

(`double`

) – The exponent of the power cone. Must be between 0 and 1.`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.`alphas`

(`double`

[]) – The weights of the power cone. Must be positive.

- Return

- Domain.InPExpCone¶
ConeDomain Domain.InPExpCone() ConeDomain Domain.InPExpCone(int m) ConeDomain Domain.InPExpCone(int[] dims)

Defines the domain of primal exponential cones:

\[\left\{ x\in \real^3 ~:~ x_1 \geq x_2 e^{x_3/x_2},\ x_1,x_2> 0 \right\}\]The conic domain scales as follows. If a variable or expression constrained to an exponential cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.InPGeoMeanCone¶
ConeDomain Domain.InPGeoMeanCone() ConeDomain Domain.InPGeoMeanCone(int n) ConeDomain Domain.InPGeoMeanCone(int m, int n) ConeDomain Domain.InPGeoMeanCone(int[] dims)

Defines the domain of primal geometric mean cones:

\[\left\{ x\in \real^n ~:~ \left(\prod_{i=1}^{n-1} x_i\right)^{1/(n-1)} \geq |x_n|,\ x_1\ldots,x_{n-1}\geq 0 \right\}\]The conic domain scales as follows. If a variable or expression constrained to a cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`n`

(`int`

) – The size of each cone; at least 2.`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.InPPowerCone¶
ConeDomain Domain.InPPowerCone(double alpha) ConeDomain Domain.InPPowerCone(double alpha, int m) ConeDomain Domain.InPPowerCone(double alpha, int[] dims) ConeDomain Domain.InPPowerCone(double[] alphas) ConeDomain Domain.InPPowerCone(double[] alphas, int m) ConeDomain Domain.InPPowerCone(double[] alphas, int[] dims)

Defines the domain of primal power cones. For a single double argument

`alpha`

it defines the set\[\left\{ x\in \real^n ~:~ x_1^\alpha x_2^{1-\alpha} \geq \sqrt{\sum_{i=3}^n x_i^2},\ x_1,x_2\geq 0 \right\}.\]For an array

`alphas`

of length \(n_l\), consisting of weights for the cone, it defines the set\[\left\{ x\in\real^n~:~ \prod_{i=1}^{n_l}x_i^{\beta_i} \geq \sqrt{x_{n_l+1}^2+\cdots+x_n^2},\ x_1,\ldots,x_{n_l}\geq 0 \right\}.\]where \(\beta_i\) are the weights normalized to add up to \(1\), ie. \(\beta_i=\alpha_i/(\sum_j \alpha_j)\) for \(i=1,\ldots,n_l\).

The conic domain scales as follows. If a variable or expression constrained to a power cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`alpha`

(`double`

) – The exponent of the power cone. Must be between 0 and 1.`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.`alphas`

(`double`

[]) – The weights of the power cone. Must be positive.

- Return

- Domain.InPSDCone¶
PSDDomain Domain.InPSDCone() PSDDomain Domain.InPSDCone(int n) PSDDomain Domain.InPSDCone(int n, int m)

When used to create a new variable in

`Model.Variable`

it defines a domain of symmetric positive semidefinite matrices, that is\[\PSD^n = \left\{ X \in \real^{n\times n} ~:~ X=X^T,\ y^TXy\geq 0,\ \mbox{for all}\ y \right\}.\]The shape of the result is \(n\times n\). If \(m\) was given the domain is a product of \(m\) such cones, that is of shape \(m\times n\times n\).

When used to impose a constraint in

`Model.Constraint`

it defines a domain\[\left\{ X \in \real^{n\times n} ~:~ \half (X + X^T) \in \PSD^n \right\}.\]i.e. a positive semidefinite matrix without the symmetry assumption.

- Parameters
`n`

(`int`

) – Dimension of the PSD matrix.`m`

(`int`

) – Number of matrices (default 1).

- Return

- Domain.InQCone¶
ConeDomain Domain.InQCone() ConeDomain Domain.InQCone(int n) ConeDomain Domain.InQCone(int m, int n) ConeDomain Domain.InQCone(int[] dims)

Defines the domain of quadratic cones:

\[\left\{ x\in \real^n ~:~ x_1^2 \geq \sum_{i=2}^n x_i^2,~ x_1 \geq 0 \right\}\]The conic domain scales as follows. If a variable or expression constrained to a quadratic cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`n`

(`int`

) – The size of each cone; at least 2.`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.InRange¶
RangeDomain Domain.InRange(double lb, double ub) RangeDomain Domain.InRange(double lb, double[] uba) RangeDomain Domain.InRange(double[] lba, double ub) RangeDomain Domain.InRange(double[] lba, double[] uba) RangeDomain Domain.InRange(double lb, double ub, int[] dims) RangeDomain Domain.InRange(double lb, double[] uba, int[] dims) RangeDomain Domain.InRange(double[] lba, double ub, int[] dims) RangeDomain Domain.InRange(double[] lba, double[] uba, int[] dims) RangeDomain Domain.InRange(double[,] lba, double[,] uba) RangeDomain Domain.InRange(Matrix lbm, Matrix ubm)

Creates a domain specified by a range in each dimension.

- Parameters
`lb`

(`double`

) – The lower bound as a common scalar value.`ub`

(`double`

) – The upper bound as a common scalar value.`uba`

(`double`

[]) – The upper bounds as an array.`uba`

(`double`

[,]) – The upper bounds as an array.`lba`

(`double`

[]) – The lower bounds as an array.`lba`

(`double`

[,]) – The lower bounds as an array.`dims`

(`int`

[]) – A list of dimension sizes.

- Return

- Domain.InRotatedQCone¶
ConeDomain Domain.InRotatedQCone() ConeDomain Domain.InRotatedQCone(int n) ConeDomain Domain.InRotatedQCone(int m, int n) ConeDomain Domain.InRotatedQCone(int[] dims)

Defines the domain of rotated quadratic cones:

\[\left\{ x\in\real^n ~:~ 2x_1 x_2 \geq \sum_{i=3}^n x_i^2,~ x_1,x_2 \geq 0 \right\}\]The conic domain scales as follows. If a variable or expression constrained to a quadratic cone is not a single vector but a \(d\)-dimensional variable then the conic domain is applicable to all vectors obtained by fixing the first \(d-1\) coordinates and moving along the last coordinate. If \(d=2\) it means that each row of a matrix must belong to a cone. See also

`Domain.Axis`

.If \(m\) was given the domain is a product of \(m\) such cones.

- Parameters
`n`

(`int`

) – The size of each cone; at least 3.`m`

(`int`

) – The number of cones (default 1).`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.InSVecPSDCone¶
ConeDomain Domain.InSVecPSDCone() ConeDomain Domain.InSVecPSDCone(int n) ConeDomain Domain.InSVecPSDCone(int d1, int d2) ConeDomain Domain.InSVecPSDCone(int[] dims)

Creates a domain of vectorized Positive Semidefinite matrices:

\[\{(x_1,\ldots,x_{d(d+1)/2})\in \real^n~:~ \mathrm{sMat}(x)\in\PSD^d\} = \{\mathrm{sVec}(X)~:~X\in\PSD^d\},\]where

\[\mathrm{sVec}(X) = (X_{11},\sqrt{2}X_{21},\ldots,\sqrt{2}X_{d1},X_{22},\sqrt{2}X_{32},\ldots,X_{dd}),\]and

\[\begin{split}\mathrm{sMat}(x) = \left[\begin{array}{cccc}x_1 & x_2/\sqrt{2} & \cdots & x_{d}/\sqrt{2} \\ x_2/\sqrt{2} & x_{d+1} & \cdots & x_{2d-1}/\sqrt{2} \\ \cdots & \cdots & \cdots & \cdots \\ x_{d}/\sqrt{2} & x_{2d-1}/\sqrt{2} & \cdots & x_{d(d+1)/2}\end{array}\right].\end{split}\]In other words, the domain consists of vectorizations of the lower-triangular part of a positive semidefinite matrix, with the non-diagonal elements additionally rescaled.

- Parameters
`n`

(`int`

) – Length of the vectorization - this must be of the form \(d*(d+1)/2\) for some positive integer \(d\).`d1`

(`int`

) – Size of first dimension of the domain.`d2`

(`int`

) – Size of second dimension of the domain.`dims`

(`int`

[]) – Shape of the domain.

- Return

- Domain.Integral¶
ConeDomain Domain.Integral(ConeDomain c) LinearDomain Domain.Integral(LinearDomain ld) RangeDomain Domain.Integral(RangeDomain rd)

Modify a given domain restricting its elements to be integral. An integral domain can only be used when creating variables, but is not allowed in a constraint. Another way of restricting variables to be integers is the method

`Variable.MakeInteger`

.- Parameters
`c`

(`ConeDomain`

) – A conic domain.`ld`

(`LinearDomain`

) – A linear domain.`rd`

(`RangeDomain`

) – A ranged domain.

- Return

- Domain.IsTrilPSD¶
PSDDomain Domain.IsTrilPSD() PSDDomain Domain.IsTrilPSD(int n) PSDDomain Domain.IsTrilPSD(int n, int m)

Creates an object representing a cone of the form

\[\left\{ X \in \real^{n\times n} ~:~ \mbox{tril}(X) \in \PSD^n \right\}.\]i.e. the lower triangular part of \(X\) defines the symmetric matrix that is positive semidefinite. The shape of the result is \(n\times n\). If \(m\) was given the domain is a product of \(m\) such cones, that is of shape \(m\times n\times n\).

- Parameters
`n`

(`int`

) – Dimension of the PSD matrix.`m`

(`int`

) – Number of matrices (default 1).

- Return

- Domain.LessThan¶
LinearDomain Domain.LessThan(double b) LinearDomain Domain.LessThan(double b, int n) LinearDomain Domain.LessThan(double b, int m, int n) LinearDomain Domain.LessThan(double b, int[] dims) LinearDomain Domain.LessThan(double[] a1) LinearDomain Domain.LessThan(double[,] a2) LinearDomain Domain.LessThan(double[] a1, int[] dims) LinearDomain Domain.LessThan(Matrix mx)

Defines the domain specified by an upper bound in each dimension.

- Parameters
`b`

(`double`

) – A single value. This is scalable: it means that each element in the variable or constraint is less than or equal to \(b\).`n`

(`int`

) – Dimension size.`m`

(`int`

) – Dimension size.`dims`

(`int`

[]) – A list of dimension sizes.`a1`

(`double`

[]) – A one-dimensional array of bounds. The shape must match the variable or constraint with which it is used.`a2`

(`double`

[,]) – A two-dimensional array of bounds. The shape must match the variable or constraint with which it is used.`mx`

(`Matrix`

) – A matrix of bound values. The shape must match the variable or constraint with which it is used.

- Return

- Domain.Sparse¶
LinearDomain Domain.Sparse(LinearDomain ld, int[] sparsity) LinearDomain Domain.Sparse(LinearDomain ld, int[,] sparsity) RangeDomain Domain.Sparse(RangeDomain rd, int[] sparsity) RangeDomain Domain.Sparse(RangeDomain rd, int[,] sparsity)

Given a linear domain, this method explicitly suggest to

*Fusion*that a sparse representation is helpful.- Parameters
`ld`

(`LinearDomain`

) – The linear sparse domain.`sparsity`

(`int`

[]) – Sparsity pattern.`sparsity`

(`int`

[,]) – Sparsity pattern.`rd`

(`RangeDomain`

) – The ranged sparse domain.

- Return

- Domain.Symmetric¶
SymmetricLinearDomain Domain.Symmetric(LinearDomain ld) SymmetricRangeDomain Domain.Symmetric(RangeDomain rd)

Given a linear domain \(D\) whose shape is that of square matrices, this method returns a domain consisting of symmetric matrices in \(D\).

- Parameters
`ld`

(`LinearDomain`

) – The linear domain to be symmetrized.`rd`

(`RangeDomain`

) – The ranged domain to be symmetrized.

- Return

- Domain.Unbounded¶
LinearDomain Domain.Unbounded() LinearDomain Domain.Unbounded(int n) LinearDomain Domain.Unbounded(int m, int n) LinearDomain Domain.Unbounded(int[] dims)

Creates a domain in which variables are unbounded.

- Parameters
`n`

(`int`

) – Dimension size.`m`

(`int`

) – Dimension size.`dims`

(`int`

[]) – A list of dimension sizes.

- Return