# 14.2.11 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:
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.

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

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

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:
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:
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: