14.2.9 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.inDPowerCone – Defines the dual power cone.

Domain.inPExpCone – Defines the primal exponential cone.

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.integral – Creates a domain of integral variables.

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

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.

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.

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

Defines the domain of dual power cones:

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

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.

  • m (int) – The number of cones (default 1).

  • dims (int[]) – Shape of the domain.

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

Defines the domain of primal power cones:

\[\left\{ x\in \real^n ~:~ x_1^\alpha x_2^{1-\alpha} \geq \sqrt{\sum_{i=3}^n x_i^2},\ x_1,x_2\geq 0 \right\}\]

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.

  • m (int) – The number of cones (default 1).

  • dims (int[]) – Shape of the domain.

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.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
Return
Domain.isLinPSD
Domain.isLinPSD() -> LinPSDDomain
Domain.isLinPSD(int n) -> LinPSDDomain
Domain.isLinPSD(int n, int m) -> LinPSDDomain

Creates an domain of vectors of length \(\half n(n+1)\) which are the flattenings of the lower-triangular part of a symmetric positive-semidefinite matrix \(X\). The shape of the result is \(\half n(n+1)\). If \(m\) was given the domain is a product of \(m\) such domains, that is of shape \(m \times \half n(n+1)\).

Parameters
  • n (int) – Dimension of the PSD matrix.

  • m (int) – Number of matrices (default 1).

Return

(LinPSDDomain)

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.

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
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
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)