# 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: