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

Domain.axis(ConeDomain c, int a) -> ConeDomain

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

(ConeDomain)

Domain.binary

Domain.binary(int n) -> RangeDomain
Domain.binary(int m, int n) -> RangeDomain
Domain.binary(int[] dims) -> RangeDomain
Domain.binary() -> RangeDomain

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

(RangeDomain)

Domain.equalsTo (==)

Domain.equalsTo(float b) -> LinearDomain
Domain.equalsTo(float b, int n) -> LinearDomain
Domain.equalsTo(float b, int m, int n) -> LinearDomain
Domain.equalsTo(float b, int[] dims) -> LinearDomain
Domain.equalsTo(float[] a1) -> LinearDomain
Domain.equalsTo(float[][] a2) -> LinearDomain
Domain.equalsTo(float[] a1, int[] dims) -> LinearDomain
Domain.equalsTo(Matrix mx) -> LinearDomain

Defines the domain consisting of a fixed point.

The standard Python operator == can also be used for this purpose. See Sec. 14.1.3 (Pythonic extensions) for details.

Parameters

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

• a2 (float[][]) – 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

(LinearDomain)

Domain.greaterThan (>=)

Domain.greaterThan(float b) -> LinearDomain
Domain.greaterThan(float b, int n) -> LinearDomain
Domain.greaterThan(float b, int m, int n) -> LinearDomain
Domain.greaterThan(float b, int[] dims) -> LinearDomain
Domain.greaterThan(float[] a1) -> LinearDomain
Domain.greaterThan(float[][] a2) -> LinearDomain
Domain.greaterThan(float[] a1, int[] dims) -> LinearDomain
Domain.greaterThan(Matrix mx) -> LinearDomain

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

The standard Python operator >= can also be used for this purpose. See Sec. 14.1.3 (Pythonic extensions) for details.

Parameters

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

• a2 (float[][]) – 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

(LinearDomain)

Domain.inDExpCone

Domain.inDExpCone() -> ConeDomain
Domain.inDExpCone(int m) -> ConeDomain
Domain.inDExpCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.inDGeoMeanCone

Domain.inDGeoMeanCone() -> ConeDomain
Domain.inDGeoMeanCone(int n) -> ConeDomain
Domain.inDGeoMeanCone(int m, int n) -> ConeDomain
Domain.inDGeoMeanCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.inDPowerCone

Domain.inDPowerCone(float alpha) -> ConeDomain
Domain.inDPowerCone(float alpha, int m) -> ConeDomain
Domain.inDPowerCone(float alpha, int[] dims) -> ConeDomain
Domain.inDPowerCone(float[] alphas) -> ConeDomain
Domain.inDPowerCone(float[] alphas, int m) -> ConeDomain
Domain.inDPowerCone(float[] alphas, int[] dims) -> ConeDomain

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

Return

(ConeDomain)

Domain.inPExpCone

Domain.inPExpCone() -> ConeDomain
Domain.inPExpCone(int m) -> ConeDomain
Domain.inPExpCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.inPGeoMeanCone

Domain.inPGeoMeanCone() -> ConeDomain
Domain.inPGeoMeanCone(int n) -> ConeDomain
Domain.inPGeoMeanCone(int m, int n) -> ConeDomain
Domain.inPGeoMeanCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.inPPowerCone

Domain.inPPowerCone(float alpha) -> ConeDomain
Domain.inPPowerCone(float alpha, int m) -> ConeDomain
Domain.inPPowerCone(float alpha, int[] dims) -> ConeDomain
Domain.inPPowerCone(float[] alphas) -> ConeDomain
Domain.inPPowerCone(float[] alphas, int m) -> ConeDomain
Domain.inPPowerCone(float[] alphas, int[] dims) -> ConeDomain

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

Return

(ConeDomain)

Domain.inPSDCone

Domain.inPSDCone() -> PSDDomain
Domain.inPSDCone(int n) -> PSDDomain
Domain.inPSDCone(int n, int m) -> PSDDomain

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

(PSDDomain)

Domain.inQCone

Domain.inQCone() -> ConeDomain
Domain.inQCone(int n) -> ConeDomain
Domain.inQCone(int m, int n) -> ConeDomain
Domain.inQCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.inRange

Domain.inRange(float lb, float ub) -> RangeDomain
Domain.inRange(float lb, float[] uba) -> RangeDomain
Domain.inRange(float[] lba, float ub) -> RangeDomain
Domain.inRange(float[] lba, float[] uba) -> RangeDomain
Domain.inRange(float lb, float ub, int[] dims) -> RangeDomain
Domain.inRange(float lb, float[] uba, int[] dims) -> RangeDomain
Domain.inRange(float[] lba, float ub, int[] dims) -> RangeDomain
Domain.inRange(float[] lba, float[] uba, int[] dims) -> RangeDomain
Domain.inRange(float[][] lba, float[][] uba) -> RangeDomain
Domain.inRange(Matrix lbm, Matrix ubm) -> RangeDomain

Creates a domain specified by a range in each dimension.

Parameters

• lb (float) – The lower bound as a common scalar value.

• ub (float) – The upper bound as a common scalar value.

• uba (float[]) – The upper bounds as an array.

• uba (float[][]) – The upper bounds as an array.

• lba (float[]) – The lower bounds as an array.

• lba (float[][]) – The lower bounds as an array.

• dims (int[]) – A list of dimension sizes.

• lbm (Matrix) – The lower bounds as a Matrix object.

• ubm (Matrix) – The upper bounds as a Matrix object.

Return

(RangeDomain)

Domain.inRotatedQCone

Domain.inRotatedQCone() -> ConeDomain
Domain.inRotatedQCone(int n) -> ConeDomain
Domain.inRotatedQCone(int m, int n) -> ConeDomain
Domain.inRotatedQCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.inSVecPSDCone

Domain.inSVecPSDCone() -> ConeDomain
Domain.inSVecPSDCone(int n) -> ConeDomain
Domain.inSVecPSDCone(int d1, int d2) -> ConeDomain
Domain.inSVecPSDCone(int[] dims) -> ConeDomain

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

(ConeDomain)

Domain.integral

Domain.integral(ConeDomain c) -> ConeDomain
Domain.integral(LinearDomain ld) -> LinearDomain
Domain.integral(RangeDomain rd) -> RangeDomain

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

• (ConeDomain)

• (LinearDomain)

• (RangeDomain)

Domain.isTrilPSD

Domain.isTrilPSD() -> PSDDomain
Domain.isTrilPSD(int n) -> PSDDomain
Domain.isTrilPSD(int n, int m) -> PSDDomain

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

(PSDDomain)

Domain.lessThan (<=)

Domain.lessThan(float b) -> LinearDomain
Domain.lessThan(float b, int n) -> LinearDomain
Domain.lessThan(float b, int m, int n) -> LinearDomain
Domain.lessThan(float b, int[] dims) -> LinearDomain
Domain.lessThan(float[] a1) -> LinearDomain
Domain.lessThan(float[][] a2) -> LinearDomain
Domain.lessThan(float[] a1, int[] dims) -> LinearDomain
Domain.lessThan(Matrix mx) -> LinearDomain

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

The standard Python operator <= can also be used for this purpose. See Sec. 14.1.3 (Pythonic extensions) for details.

Parameters

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

• a2 (float[][]) – 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

(LinearDomain)

Domain.sparse

Domain.sparse(LinearDomain ld, int[] sparsity) -> LinearDomain
Domain.sparse(LinearDomain ld, int[][] sparsity) -> LinearDomain
Domain.sparse(RangeDomain rd, int[] sparsity) -> RangeDomain
Domain.sparse(RangeDomain rd, int[][] sparsity) -> RangeDomain

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

• (LinearDomain)

• (RangeDomain)

Domain.symmetric

Domain.symmetric(LinearDomain ld) -> SymmetricLinearDomain
Domain.symmetric(RangeDomain rd) -> SymmetricRangeDomain

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

• (SymmetricLinearDomain)

• (SymmetricRangeDomain)

Domain.unbounded

Domain.unbounded() -> LinearDomain
Domain.unbounded(int n) -> LinearDomain
Domain.unbounded(int m, int n) -> LinearDomain
Domain.unbounded(int[] dims) -> LinearDomain

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

(LinearDomain)