Standard form

AbstractFunction

Abstract supertype for function objects.

Required methods

All functions must implement:

• Base.copy
• Base.isapprox
• constant

Abstract subtypes of AbstractFunction may require additional methods to be implemented.

output_dimension(f::AbstractFunction)

Return 1 if f is an AbstractScalarFunction, or the number of output components if f is an AbstractVectorFunction.

constant(f::AbstractFunction[, ::Type{T}]) where {T}

Returns the constant term of a scalar-valued function, or the constant vector of a vector-valued function.

If f is untyped and T is provided, returns zero(T).

constant(set::Union{EqualTo,GreaterThan,LessThan,Parameter})

Returns the constant term of the set set.

Example

julia> MOI.constant(MOI.GreaterThan(1.0))
1.0

julia> MOI.constant(MOI.LessThan(2.5))
2.5

julia> MOI.constant(MOI.EqualTo(3))
3

julia> MOI.constant(MOI.Parameter(4.5))
4.5

abstract type AbstractScalarFunction <: AbstractFunction

Abstract supertype for scalar-valued AbstractFunctions.

VariableIndex

A type-safe wrapper for Int64 for use in referencing variables in a model. To allow for deletion, indices need not be consecutive.

ScalarAffineTerm{T}(coefficient::T, variable::VariableIndex) where {T}

Represents the scalar-valued term coefficient * variable.

Example

julia> x = MOI.VariableIndex(1)
MOI.VariableIndex(1)

julia> MOI.ScalarAffineTerm(2.0, x)
MathOptInterface.ScalarAffineTerm{Float64}(2.0, MOI.VariableIndex(1))

ScalarAffineFunction{T}(
    terms::Vector{ScalarAffineTerm{T}},
    constant::T,
) where {T}

Represents the scalar-valued affine function $a^\top x + b$, where:

• $a^\top x$ is represented by the vector of ScalarAffineTerms
• $b$ is a scalar constant::T

Duplicates

Duplicate variable indices in terms are accepted, and the corresponding coefficients are summed together.

Example

julia> x = MOI.VariableIndex(1)
MOI.VariableIndex(1)

julia> terms = [MOI.ScalarAffineTerm(2.0, x), MOI.ScalarAffineTerm(3.0, x)]
2-element Vector{MathOptInterface.ScalarAffineTerm{Float64}}:
 MathOptInterface.ScalarAffineTerm{Float64}(2.0, MOI.VariableIndex(1))
 MathOptInterface.ScalarAffineTerm{Float64}(3.0, MOI.VariableIndex(1))

julia> f = MOI.ScalarAffineFunction(terms, 4.0)
4.0 + 2.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(1)

ScalarQuadraticTerm{T}(
    coefficient::T,
    variable_1::VariableIndex,
    variable_2::VariableIndex,
) where {T}

Represents the scalar-valued term $c x_i x_j$ where $c$ is coefficient, $x_i$ is variable_1 and $x_j$ is variable_2.

Example

julia> x = MOI.VariableIndex(1)
MOI.VariableIndex(1)

julia> MOI.ScalarQuadraticTerm(2.0, x, x)
MathOptInterface.ScalarQuadraticTerm{Float64}(2.0, MOI.VariableIndex(1), MOI.VariableIndex(1))

ScalarQuadraticFunction{T}(
    quadratic_terms::Vector{ScalarQuadraticTerm{T}},
    affine_terms::Vector{ScalarAffineTerm{T}},
    constant::T,
) wher {T}

The scalar-valued quadratic function $\frac{1}{2}x^\top Q x + a^\top x + b$, where:

• $Q$ is the symmetric matrix given by the vector of ScalarQuadraticTerms
• $a^\top x$ is a sparse vector given by the vector of ScalarAffineTerms
• $b$ is the scalar constant::T.

Duplicates

Duplicate indices in quadratic_terms or affine_terms are accepted, and the corresponding coefficients are summed together.

In quadratic_terms, "mirrored" indices, (q, r) and (r, q) where r and q are VariableIndexes, are considered duplicates; only one needs to be specified.

The 0.5 factor

Coupled with the interpretation of mirrored indices, the 0.5 factor in front of the $Q$ matrix is a common source of bugs.

As a rule, to represent $a * x^2 + b * x * y$:

• The coefficient $a$ in front of squared variables (diagonal elements in $Q$) must be doubled when creating a ScalarQuadraticTerm
• The coefficient $b$ in front of off-diagonal elements in $Q$ should be left as $b$, be cause the mirrored index $b * y * x$ will be implicitly added.

Example

To represent the function $f(x, y) = 2 * x^2 + 3 * x * y + 4 * x + 5$, do:

julia> x = MOI.VariableIndex(1);

julia> y = MOI.VariableIndex(2);

julia> constant = 5.0;

julia> affine_terms = [MOI.ScalarAffineTerm(4.0, x)];

julia> quadratic_terms = [
           MOI.ScalarQuadraticTerm(4.0, x, x),  # Note the changed coefficient
           MOI.ScalarQuadraticTerm(3.0, x, y),
       ]
2-element Vector{MathOptInterface.ScalarQuadraticTerm{Float64}}:
 MathOptInterface.ScalarQuadraticTerm{Float64}(4.0, MOI.VariableIndex(1), MOI.VariableIndex(1))
 MathOptInterface.ScalarQuadraticTerm{Float64}(3.0, MOI.VariableIndex(1), MOI.VariableIndex(2))

julia> f = MOI.ScalarQuadraticFunction(quadratic_terms, affine_terms, constant)
5.0 + 4.0 MOI.VariableIndex(1) + 2.0 MOI.VariableIndex(1)² + 3.0 MOI.VariableIndex(1)*MOI.VariableIndex(2)

ScalarNonlinearFunction(head::Symbol, args::Vector{Any})

The scalar-valued nonlinear function head(args...), represented as a symbolic expression tree, with the call operator head and ordered arguments in args.

head

The head::Symbol must be an operator supported by the model.

The default list of supported univariate operators is given by:

• Nonlinear.DEFAULT_UNIVARIATE_OPERATORS

and the default list of supported multivariate operators is given by:

• Nonlinear.DEFAULT_MULTIVARIATE_OPERATORS

Additional operators can be registered by setting a UserDefinedFunction attribute.

See the full list of operators supported by a ModelLike by querying ListOfSupportedNonlinearOperators.

args

The vector args contains the arguments to the nonlinear function. If the operator is univariate, it must contain one element. Otherwise, it may contain multiple elements.

Each element must be one of the following:

• A constant value of type T<:Real
• A VariableIndex
• A ScalarAffineFunction
• A ScalarQuadraticFunction
• A ScalarNonlinearFunction

Unsupported operators

If the optimizer does not support head, an UnsupportedNonlinearOperator error will be thrown.

There is no guarantee about when this error will be thrown; it may be thrown when the function is first added to the model, or it may be thrown when optimize! is called.

Example

To represent the function $f(x) = sin(x)^2$, do:

julia> x = MOI.VariableIndex(1)
MOI.VariableIndex(1)

julia> MOI.ScalarNonlinearFunction(
           :^,
           Any[MOI.ScalarNonlinearFunction(:sin, Any[x]), 2],
       )
^(sin(MOI.VariableIndex(1)), (2))

abstract type AbstractVectorFunction <: AbstractFunction

Abstract supertype for vector-valued AbstractFunctions.

Required methods

All subtypes of AbstractVectorFunction must implement:

• output_dimension

VectorOfVariables(variables::Vector{VariableIndex}) <: AbstractVectorFunction

The vector-valued function f(x) = variables, where variables is a subset of VariableIndexes in the model.

The list of variables may contain duplicates.

Example

julia> x = MOI.VariableIndex.(1:2)
2-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)

julia> f = MOI.VectorOfVariables([x[1], x[2], x[1]])
┌                    ┐
│MOI.VariableIndex(1)│
│MOI.VariableIndex(2)│
│MOI.VariableIndex(1)│
└                    ┘

julia> MOI.output_dimension(f)
3

VectorAffineTerm{T}(
    output_index::Int64,
    scalar_term::ScalarAffineTerm{T},
) where {T}

A VectorAffineTerm is a scalar_term that appears in the output_index row of the vector-valued VectorAffineFunction or VectorQuadraticFunction.

Example

julia> x = MOI.VariableIndex(1);

julia> MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x))
MathOptInterface.VectorAffineTerm{Float64}(2, MathOptInterface.ScalarAffineTerm{Float64}(3.0, MOI.VariableIndex(1)))

VectorAffineFunction{T}(
    terms::Vector{VectorAffineTerm{T}},
    constants::Vector{T},
) where {T}

The vector-valued affine function $A x + b$, where:

• $A x$ is the sparse matrix given by the vector of VectorAffineTerms
• $b$ is the vector constants

Duplicates

Duplicate indices in the $A$ are accepted, and the corresponding coefficients are summed together.

Example

julia> x = MOI.VariableIndex(1);

julia> terms = [
           MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),
           MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x)),
       ];

julia> f = MOI.VectorAffineFunction(terms, [4.0, 5.0])
┌                              ┐
│4.0 + 2.0 MOI.VariableIndex(1)│
│5.0 + 3.0 MOI.VariableIndex(1)│
└                              ┘

julia> MOI.output_dimension(f)
2

VectorQuadraticTerm{T}(
    output_index::Int64,
    scalar_term::ScalarQuadraticTerm{T},
) where {T}

A VectorQuadraticTerm is a ScalarQuadraticTerm scalar_term that appears in the output_index row of the vector-valued VectorQuadraticFunction.

Example

julia> x = MOI.VariableIndex(1);

julia> MOI.VectorQuadraticTerm(Int64(2), MOI.ScalarQuadraticTerm(3.0, x, x))
MathOptInterface.VectorQuadraticTerm{Float64}(2, MathOptInterface.ScalarQuadraticTerm{Float64}(3.0, MOI.VariableIndex(1), MOI.VariableIndex(1)))

VectorQuadraticFunction{T}(
    quadratic_terms::Vector{VectorQuadraticTerm{T}},
    affine_terms::Vector{VectorAffineTerm{T}},
    constants::Vector{T},
) where {T}

The vector-valued quadratic function with ith component ("output index") defined as $\frac{1}{2}x^\top Q_i x + a_i^\top x + b_i$, where:

• $\frac{1}{2}x^\top Q_i x$ is the symmetric matrix given by the VectorQuadraticTerm elements in quadratic_terms with output_index == i
• $a_i^\top x$ is the sparse vector given by the VectorAffineTerm elements in affine_terms with output_index == i
• $b_i$ is a scalar given by constants[i]

Duplicates

Duplicate indices in quadratic_terms and affine_terms with the same output_index are handled in the same manner as duplicates in ScalarQuadraticFunction.

Example

julia> x = MOI.VariableIndex(1);

julia> y = MOI.VariableIndex(2);

julia> constants = [4.0, 5.0];

julia> affine_terms = [
           MOI.VectorAffineTerm(Int64(1), MOI.ScalarAffineTerm(2.0, x)),
           MOI.VectorAffineTerm(Int64(2), MOI.ScalarAffineTerm(3.0, x)),
       ];

julia> quad_terms = [
        MOI.VectorQuadraticTerm(Int64(1), MOI.ScalarQuadraticTerm(2.0, x, x)),
        MOI.VectorQuadraticTerm(Int64(2), MOI.ScalarQuadraticTerm(3.0, x, y)),
           ];

julia> f = MOI.VectorQuadraticFunction(quad_terms, affine_terms, constants)
┌                                                                              ┐
│4.0 + 2.0 MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(1)²                    │
│5.0 + 3.0 MOI.VariableIndex(1) + 3.0 MOI.VariableIndex(1)*MOI.VariableIndex(2)│
└                                                                              ┘

julia> MOI.output_dimension(f)
2

VectorNonlinearFunction(args::Vector{ScalarNonlinearFunction})

The vector-valued nonlinear function composed of a vector of ScalarNonlinearFunction.

args

The vector args contains the scalar elements of the nonlinear function. Each element must be a ScalarNonlinearFunction, but if you pass a Vector{Any}, the elements can be automatically converted from one of the following:

• A constant value of type T<:Real
• A VariableIndex
• A ScalarAffineFunction
• A ScalarQuadraticFunction
• A ScalarNonlinearFunction

Example

To represent the function $f(x) = [sin(x)^2, x]$, do:

julia> x = MOI.VariableIndex(1)
MOI.VariableIndex(1)

julia> g = MOI.ScalarNonlinearFunction(
           :^,
           Any[MOI.ScalarNonlinearFunction(:sin, Any[x]), 2.0],
       )
^(sin(MOI.VariableIndex(1)), 2.0)

julia> MOI.VectorNonlinearFunction([g, x])
┌                                 ┐
│^(sin(MOI.VariableIndex(1)), 2.0)│
│+(MOI.VariableIndex(1))          │
└                                 ┘

Note the automatic conversion from x to +(x).

AbstractSet

Abstract supertype for set objects used to encode constraints.

Required methods

For sets of type S with isbitstype(S) == false, you must implement:

• Base.copy(set::S)
• Base.:(==)(x::S, y::S)

Subtypes of AbstractSet such as AbstractScalarSet and AbstractVectorSet may prescribe additional required methods.

Optional methods

You may optionally implement:

• dual_set
• dual_set_type

Note for developers

When creating a new set, the set struct must not contain any VariableIndex or ConstraintIndex objects.

AbstractScalarSet

Abstract supertype for subsets of $\mathbb{R}$.

AbstractVectorSet

Abstract supertype for subsets of $\mathbb{R}^n$ for some $n$.

Required methods

All AbstractVectorSets of type S must implement:

• dimension, unless the dimension is stored in the set.dimension field
• Utilities.set_dot, unless the dot product between two vectors in the set is equivalent to LinearAlgebra.dot.

dimension(set::AbstractSet)

Return the output_dimension that an AbstractFunction should have to be used with the set set.

Example

julia> MOI.dimension(MOI.Reals(4))
4

julia> MOI.dimension(MOI.LessThan(3.0))
1

julia> MOI.dimension(MOI.PositiveSemidefiniteConeTriangle(2))
3

dual_set(set::AbstractSet)

Return the dual set of set, that is the dual cone of the set. This follows the definition of duality discussed in Duality.

See Dual cone for more information.

If the dual cone is not defined it returns an error.

Example

julia> MOI.dual_set(MOI.Reals(4))
MathOptInterface.Zeros(4)

julia> MOI.dual_set(MOI.SecondOrderCone(5))
MathOptInterface.SecondOrderCone(5)

julia> MOI.dual_set(MOI.ExponentialCone())
MathOptInterface.DualExponentialCone()

dual_set_type(S::Type{<:AbstractSet})

Return the type of dual set of sets of type S, as returned by dual_set. If the dual cone is not defined it returns an error.

Example

julia> MOI.dual_set_type(MOI.Reals)
MathOptInterface.Zeros

julia> MOI.dual_set_type(MOI.SecondOrderCone)
MathOptInterface.SecondOrderCone

julia> MOI.dual_set_type(MOI.ExponentialCone)
MathOptInterface.DualExponentialCone

constant(set::Union{EqualTo,GreaterThan,LessThan,Parameter})

Returns the constant term of the set set.

Example

julia> MOI.constant(MOI.GreaterThan(1.0))
1.0

julia> MOI.constant(MOI.LessThan(2.5))
2.5

julia> MOI.constant(MOI.EqualTo(3))
3

julia> MOI.constant(MOI.Parameter(4.5))
4.5

supports_dimension_update(S::Type{<:MOI.AbstractVectorSet})

Return a Bool indicating whether the elimination of any dimension of n-dimensional sets of type S give an n-1-dimensional set S. By default, this function returns false so it should only be implemented for sets that supports dimension update.

For instance, supports_dimension_update(MOI.Nonnegatives) is true because the elimination of any dimension of the n-dimensional nonnegative orthant gives the n-1-dimensional nonnegative orthant. However supports_dimension_update(MOI.ExponentialCone) is false.

update_dimension(s::AbstractVectorSet, new_dim::Int)

Returns a set with the dimension modified to new_dim.

GreaterThan{T<:Real}(lower::T)

The set $[lower, \infty) \subseteq \mathbb{R}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.GreaterThan(0.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.GreaterThan{Float64}}(1)

LessThan{T<:Real}(upper::T)

The set $(-\infty, upper] \subseteq \mathbb{R}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.LessThan(2.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.LessThan{Float64}}(1)

EqualTo{T<:Number}(value::T)

The set containing the single point $\{value\} \subseteq \mathbb{R}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.EqualTo(2.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.EqualTo{Float64}}(1)

Interval{T<:Real}(lower::T, upper::T)

The interval $[lower, upper] \subseteq \mathbb{R} \cup \{-\infty, +\infty\}$.

If lower or upper is -Inf or Inf, respectively, the set is interpreted as a one-sided interval.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.Interval(1.0, 2.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)

Integer()

The set of integers, $\mathbb{Z}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.Integer())
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1)

ZeroOne()

The set $\{0, 1\}$.

Variables belonging to the ZeroOne set are also known as "binary" variables.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.ZeroOne())
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1)

Semicontinuous{T<:Real}(lower::T, upper::T)

The set $\{0\} \cup [lower, upper]$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.Semicontinuous(2.0, 3.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Semicontinuous{Float64}}(1)

Semiinteger{T<:Real}(lower::T, upper::T)

The set $\{0\} \cup \{lower, lower+1, \ldots, upper-1, upper\}$.

Note that if lower and upper are not equivalent to an integer, then the solver may throw an error, or it may round up lower and round down upper to the nearest integers.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> MOI.add_constraint(model, x, MOI.Semiinteger(2.0, 3.0))
MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Semiinteger{Float64}}(1)

Parameter{T<:Number}(value::T)

The set containing the single point $\{value\} \subseteq \mathbb{R}$.

The Parameter set is conceptually similar to the EqualTo set, except that a variable constrained to the Parameter set cannot have other constraints added to it, and the Parameter set can never be deleted. Thus, solvers are free to treat the variable as a constant, and they need not add it as a decision variable to the model.

Because of this behavior, you must add parameters using add_constrained_variable, and solvers should declare supports_add_constrained_variable and not supports_constraint for the Parameter set.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> p, ci = MOI.add_constrained_variable(model, MOI.Parameter(2.5))
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Parameter{Float64}}(1))

julia> MOI.set(model, MOI.ConstraintSet(), ci, MOI.Parameter(3.0))

julia> MOI.get(model, MOI.ConstraintSet(), ci)
MathOptInterface.Parameter{Float64}(3.0)

Reals(dimension::Int)

The set $\mathbb{R}^{dimension}$ (containing all points) of non-negative dimension dimension.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Reals(3))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Reals}(1)

Zeros(dimension::Int)

The set $\{ 0 \}^{dimension}$ (containing only the origin) of non-negative dimension dimension.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Zeros(3))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Zeros}(1)

Nonnegatives(dimension::Int)

The nonnegative orthant $\{ x \in \mathbb{R}^{dimension} : x \ge 0 \}$ of non-negative dimension dimension.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonnegatives(3))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)

Nonpositives(dimension::Int)

The nonpositive orthant $\{ x \in \mathbb{R}^{dimension} : x \le 0 \}$ of non-negative dimension dimension.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Nonpositives(3))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonpositives}(1)

NormInfinityCone(dimension::Int)

The $\ell_\infty$-norm cone $\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \lVert x \rVert_\infty = \max_i \lvert x_i \rvert \}$ of dimension dimension.

The dimension must be at least 1.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormInfinityCone(4))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormInfinityCone}(1)

NormOneCone(dimension::Int)

The $\ell_1$-norm cone $\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \lVert x \rVert_1 = \sum_i \lvert x_i \rvert \}$ of dimension dimension.

The dimension must be at least 1.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormOneCone(4))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormOneCone}(1)

NormCone(p::Float64, dimension::Int)

The $\ell_p$-norm cone $\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \left(\sum\limits_i |x_i|^p\right)^{\frac{1}{p}} \}$ of dimension dimension.

The dimension must be at least 1.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.NormCone(3, 4))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormCone}(1)

SecondOrderCone(dimension::Int)

The second-order cone (or Lorenz cone or $\ell_2$-norm cone) $\{ (t,x) \in \mathbb{R}^{dimension} : t \ge \lVert x \rVert_2 \}$ of dimension dimension.

The dimension must be at least 1.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables([t; x]), MOI.SecondOrderCone(4))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SecondOrderCone}(1)

RotatedSecondOrderCone(dimension::Int)

The rotated second-order cone $\{ (t,u,x) \in \mathbb{R}^{dimension} : 2tu \ge \lVert x \rVert_2^2, t,u \ge 0 \}$ of dimension dimension.

The dimension must be at least 2.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> u = MOI.add_variable(model)
MOI.VariableIndex(2)

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; u; x]),
           MOI.RotatedSecondOrderCone(5),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RotatedSecondOrderCone}(1)

GeometricMeanCone(dimension::Int)

The geometric mean cone $\{ (t,x) \in \mathbb{R}^{n+1} : x \ge 0, t \le \sqrt[n]{x_1 x_2 \cdots x_n} \}$, where dimension = n + 1 >= 2.

Duality note

The dual of the geometric mean cone is $\{ (u, v) \in \mathbb{R}^{n+1} : u \le 0, v \ge 0, -u \le n \sqrt[n]{\prod_i v_i} \}$, where dimension = n + 1 >= 2.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; x]),
           MOI.GeometricMeanCone(4),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.GeometricMeanCone}(1)

ExponentialCone()

The 3-dimensional exponential cone $\{ (x,y,z) \in \mathbb{R}^3 : y \exp (x/y) \le z, y > 0 \}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.ExponentialCone())
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.ExponentialCone}(1)

DualExponentialCone()

The 3-dimensional dual exponential cone $\{ (u,v,w) \in \mathbb{R}^3 : -u \exp (v/u) \le \exp(1) w, u < 0 \}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.DualExponentialCone())
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.DualExponentialCone}(1)

PowerCone{T<:Real}(exponent::T)

The 3-dimensional power cone $\{ (x,y,z) \in \mathbb{R}^3 : x^{exponent} y^{1-exponent} \ge |z|, x \ge 0, y \ge 0 \}$ with parameter exponent.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.PowerCone(0.5))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.PowerCone{Float64}}(1)

DualPowerCone{T<:Real}(exponent::T)

The 3-dimensional power cone $\{ (u,v,w) \in \mathbb{R}^3 : (\frac{u}{exponent})^{exponent} (\frac{v}{1-exponent})^{1-exponent} \ge |w|, u \ge 0, v \ge 0 \}$ with parameter exponent.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.DualPowerCone(0.5))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.DualPowerCone{Float64}}(1)

RelativeEntropyCone(dimension::Int)

The relative entropy cone $\{ (u, v, w) \in \mathbb{R}^{1+2n} : u \ge \sum_{i=1}^n w_i \log(\frac{w_i}{v_i}), v_i \ge 0, w_i \ge 0 \}$, where dimension = 2n + 1 >= 1.

Duality note

The dual of the relative entropy cone is $\{ (u, v, w) \in \mathbb{R}^{1+2n} : \forall i, w_i \ge u (\log (\frac{u}{v_i}) - 1), v_i \ge 0, u > 0 \}$ of dimension dimension${}=2n+1$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> u = MOI.add_variable(model);

julia> v = MOI.add_variables(model, 3);

julia> w = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([u; v; w]),
           MOI.RelativeEntropyCone(7),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RelativeEntropyCone}(1)

NormSpectralCone(row_dim::Int, column_dim::Int)

The epigraph of the matrix spectral norm (maximum singular value function) $\{ (t, X) \in \mathbb{R}^{1 + row_dim \times column_dim} : t \ge \sigma_1(X) \}$, where $\sigma_i$ is the $i$th singular value of the matrix $X$ of non-negative row dimension row_dim and column dimension column_dim.

The matrix X is vectorized by stacking the columns, matching the behavior of Julia's vec function.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> X = reshape(MOI.add_variables(model, 6), 2, 3)
2×3 Matrix{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(2)  MOI.VariableIndex(4)  MOI.VariableIndex(6)
 MOI.VariableIndex(3)  MOI.VariableIndex(5)  MOI.VariableIndex(7)

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; vec(X)]),
           MOI.NormSpectralCone(2, 3),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormSpectralCone}(1)

NormNuclearCone(row_dim::Int, column_dim::Int)

The epigraph of the matrix nuclear norm (sum of singular values function) $\{ (t, X) \in \mathbb{R}^{1 + row_dim \times column_dim} : t \ge \sum_i \sigma_i(X) \}$, where $\sigma_i$ is the $i$th singular value of the matrix $X$ of non-negative row dimension row_dim and column dimension column_dim.

The matrix X is vectorized by stacking the columns, matching the behavior of Julia's vec function.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> X = reshape(MOI.add_variables(model, 6), 2, 3)
2×3 Matrix{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(2)  MOI.VariableIndex(4)  MOI.VariableIndex(6)
 MOI.VariableIndex(3)  MOI.VariableIndex(5)  MOI.VariableIndex(7)

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; vec(X)]),
           MOI.NormNuclearCone(2, 3),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.NormNuclearCone}(1)

SOS1{T<:Real}(weights::Vector{T})

The set corresponding to the Special Ordered Set (SOS) constraint of Type I.

Of the variables in the set, at most one can be nonzero.

The weights induce an ordering of the variables such that the kth element in the set corresponds to the kth weight in weights. Solvers may use these weights to improve the efficiency of the solution process, but the ordering does not change the set of feasible solutions.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables(x),
           MOI.SOS1([1.0, 3.0, 2.5]),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SOS1{Float64}}(1)

SOS2{T<:Real}(weights::Vector{T})

The set corresponding to the Special Ordered Set (SOS) constraint of Type II.

The weights induce an ordering of the variables such that the kth element in the set corresponds to the kth weight in weights. Therefore, the weights must be unique.

Of the variables in the set, at most two can be nonzero, and if two are nonzero, they must be adjacent in the ordering of the set.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables(x),
           MOI.SOS2([1.0, 3.0, 2.5]),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.SOS2{Float64}}(1)

Indicator{ACTIVATE_ON_ZERO}(set::AbstractScalarSet)
Indicator{ACTIVATE_ON_ONE}(set::AbstractScalarSet)

The set corresponding to an indicator constraint.

The type parameter must be an ActivationCondition.

When the type parameter is ACTIVATE_ON_ZERO, this means:

$\{(y, x) \in \{0, 1\} \times \mathbb{R} : y = 0 \implies x \in set\}$

When the type parameter is ACTIVATE_ON_ONE, this means:

$\{(y, x) \in \{0, 1\} \times \mathbb{R} : y = 1 \implies x \in set\}$

Notes

Most solvers expect that the first row of the function is interpretable as a VariableIndex (for example, 1.0 * x + 0.0), and that the variable is constrained to the ZeroOne set. An error will be thrown if this is not the case.

Example

The constraint $\{(y, x) \in \{0, 1\} \times \mathbb{R}^2 : y = 1 \implies x_1 + x_2 \leq 9 \}$ is defined as

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 2);

julia> y, _ = MOI.add_constrained_variable(model, MOI.ZeroOne());

julia> f = MOI.Utilities.vectorize([y, 1.0 * x[1] + 1.0 * x[2]])
┌                                                         ┐
│0.0 + 1.0 MOI.VariableIndex(3)                           │
│0.0 + 1.0 MOI.VariableIndex(1) + 1.0 MOI.VariableIndex(2)│
└                                                         ┘

julia> s = MOI.Indicator{MOI.ACTIVATE_ON_ONE}(MOI.LessThan(9.0))
MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ONE, MathOptInterface.LessThan{Float64}}(MathOptInterface.LessThan{Float64}(9.0))

julia> MOI.add_constraint(model, f, s)
MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ONE, MathOptInterface.LessThan{Float64}}}(1)

The constraint $\{(y, x) \in \{0, 1\} \times \mathbb{R} : y = 0 \implies x = 0 \}$ is defined as

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variable(model);

julia> y, _ = MOI.add_constrained_variable(model, MOI.ZeroOne());

julia> f = MOI.VectorOfVariables([y, x]);

julia> s = MOI.Indicator{MOI.ACTIVATE_ON_ZERO}(MOI.EqualTo(0.0))
MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ZERO, MathOptInterface.EqualTo{Float64}}(MathOptInterface.EqualTo{Float64}(0.0))

julia> MOI.add_constraint(model, f, s)
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Indicator{MathOptInterface.ACTIVATE_ON_ZERO, MathOptInterface.EqualTo{Float64}}}(1)

ActivationCondition

Activation condition for an indicator constraint.

The enum value is used as first type parameter of Indicator{A,S}.

Values

ACTIVATE_ON_ZERO

The indicator constraint holds when the binary variable is zero.

ACTIVATE_ON_ONE

The indicator constraint holds when the binary variable is one.

ACTIVATE_ON_ZERO::ActivationCondition

An instance of the ActivationCondition enum.

About

The indicator constraint holds when the binary variable is zero.

ACTIVATE_ON_ONE::ActivationCondition

An instance of the ActivationCondition enum.

About

The indicator constraint holds when the binary variable is one.

Complements(dimension::Base.Integer)

The set corresponding to a mixed complementarity constraint.

Complementarity constraints should be specified with an AbstractVectorFunction-in-Complements(dimension) constraint.

The dimension of the vector-valued function F must be dimension. This defines a complementarity constraint between the scalar function F[i] and the variable in F[i + dimension/2]. Thus, F[i + dimension/2] must be interpretable as a single variable x_i (for example, 1.0 * x + 0.0), and dimension must be even.

The mixed complementarity problem consists of finding x_i in the interval [lb, ub] (that is, in the set Interval(lb, ub)), such that the following holds:

1. F_i(x) == 0 if lb_i < x_i < ub_i
2. F_i(x) >= 0 if lb_i == x_i
3. F_i(x) <= 0 if x_i == ub_i

Classically, the bounding set for x_i is Interval(0, Inf), which recovers: 0 <= F_i(x) ⟂ x_i >= 0, where the ⟂ operator implies F_i(x) * x_i = 0.

Example

The problem:

x -in- Interval(-1, 1)
[-4 * x - 3, x] -in- Complements(2)

defines the mixed complementarity problem where the following holds:

1. -4 * x - 3 == 0 if -1 < x < 1
2. -4 * x - 3 >= 0 if x == -1
3. -4 * x - 3 <= 0 if x == 1

There are three solutions:

1. x = -3/4 with F(x) = 0
2. x = -1 with F(x) = 1
3. x = 1 with F(x) = -7

julia> model = MOI.Utilities.Model{Float64}();

julia> x, _ = MOI.add_constrained_variable(model, MOI.Interval(-1.0, 1.0));

julia> MOI.add_constraint(
            model,
            MOI.Utilities.vectorize([-4.0 * x - 3.0, x]),
            MOI.Complements(2),
        )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Complements}(1)

The function F can also be defined in terms of single variables. For example, the problem:

[x_3, x_4] -in- Nonnegatives(2)
[x_1, x_2, x_3, x_4] -in- Complements(4)

defines the complementarity problem where 0 <= x_1 ⟂ x_3 >= 0 and 0 <= x_2 ⟂ x_4 >= 0.

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 4);

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x[3:4]), MOI.Nonnegatives(2))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Nonnegatives}(1)

julia> MOI.add_constraint(
            model,
            MOI.VectorOfVariables(x),
            MOI.Complements(4),
        )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Complements}(1)

HyperRectangle(lower::Vector{T}, upper::Vector{T}) where {T}

The set $\{x \in \bar{\mathbb{R}}^d: x_i \in [lower_i, upper_i] \forall i=1,\ldots,d\}$.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3)
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables(x),
           MOI.HyperRectangle(zeros(3), ones(3)),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.HyperRectangle{Float64}}(1)

struct Scaled{S<:AbstractVectorSet} <: AbstractVectorSet
    set::S
end

Given a vector $a \in \mathbb{R}^d$ and a set representing the set $\mathcal{S} \in \mathbb{R}^d$ such that Utilities.set_dot for $x \in \mathcal{S}$ and $y \in \mathcal{S}^*$ is

\[\sum_{i=1}^d a_i x_i y_i\]

the set Scaled(set) is defined as

\[\{ (\sqrt{a_1} x_1, \sqrt{a_2} x_2, \ldots, \sqrt{a_d} x_d) : x \in S \}\]

Example

This can be used to scale a vector of numbers

julia> set = MOI.PositiveSemidefiniteConeTriangle(2)
MathOptInterface.PositiveSemidefiniteConeTriangle(2)

julia> a = MOI.Utilities.SetDotScalingVector{Float64}(set)
3-element MathOptInterface.Utilities.SetDotScalingVector{Float64, MathOptInterface.PositiveSemidefiniteConeTriangle}:
 1.0
 1.4142135623730951
 1.0

julia> using LinearAlgebra

julia> MOI.Utilities.operate(*, Float64, Diagonal(a), ones(3))
3-element Vector{Float64}:
 1.0
 1.4142135623730951
 1.0

It can be also used to scale a vector of function

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3);

julia> func = MOI.VectorOfVariables(x)
┌                    ┐
│MOI.VariableIndex(1)│
│MOI.VariableIndex(2)│
│MOI.VariableIndex(3)│
└                    ┘

julia> set = MOI.PositiveSemidefiniteConeTriangle(2)
MathOptInterface.PositiveSemidefiniteConeTriangle(2)

julia> MOI.Utilities.operate(*, Float64, Diagonal(a), func)
┌                                             ┐
│0.0 + 1.0 MOI.VariableIndex(1)               │
│0.0 + 1.4142135623730951 MOI.VariableIndex(2)│
│0.0 + 1.0 MOI.VariableIndex(3)               │
└                                             ┘

AllDifferent(dimension::Int)

The set $\{x \in \mathbb{Z}^{d}\}$ such that no two elements in $x$ take the same value and dimension = d.

Also known as

This constraint is called all_different in MiniZinc, and is sometimes also called distinct.

Example

To enforce x[1] != x[2] AND x[1] != x[3] AND x[2] != x[3]:

julia> model = MOI.Utilities.Model{Float64}();

julia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.AllDifferent(3))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.AllDifferent}(1)

BinPacking(c::T, w::Vector{T}) where {T}

The set $\{x \in \mathbb{Z}^d\}$ where d = length(w), such that each item i in 1:d of weight w[i] is put into bin x[i], and the total weight of each bin does not exceed c.

There are additional assumptions that the capacity, c, and the weights, w, must all be non-negative.

The bin numbers depend on the bounds of x, so they may be something other than the integers 1:d.

Also known as

This constraint is called bin_packing in MiniZinc.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> bins = MOI.add_variables(model, 5)
5-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)
 MOI.VariableIndex(4)
 MOI.VariableIndex(5)

julia> weights = Float64[1, 1, 2, 2, 3]
5-element Vector{Float64}:
 1.0
 1.0
 2.0
 2.0
 3.0

julia> MOI.add_constraint.(model, bins, MOI.Integer())
5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}}:
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(3)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(4)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(5)

julia> MOI.add_constraint.(model, bins, MOI.Interval(4.0, 6.0))
5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}}:
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(1)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(2)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(3)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(4)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Interval{Float64}}(5)

julia> MOI.add_constraint(model, MOI.VectorOfVariables(bins), MOI.BinPacking(3.0, weights))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.BinPacking{Float64}}(1)

Circuit(dimension::Int)

The set $\{x \in \{1..d\}^d\}$ that constraints $x$ to be a circuit, such that $x_i = j$ means that $j$ is the successor of $i$, and dimension = d.

Graphs with multiple independent circuits, such as [2, 1, 3] and [2, 1, 4, 3], are not valid.

Also known as

This constraint is called circuit in MiniZinc, and it is equivalent to forming a (potentially sub-optimal) tour in the travelling salesperson problem.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Circuit(3))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Circuit}(1)

CountAtLeast(n::Int, d::Vector{Int}, set::Set{Int})

The set $\{x \in \mathbb{Z}^{d_1 + d_2 + \ldots d_N}\}$, where x is partitioned into N subsets ($\{x_1, \ldots, x_{d_1}\}$, $\{x_{d_1 + 1}, \ldots, x_{d_1 + d_2}\}$ and so on), and at least $n$ elements of each subset take one of the values in set.

Also known as

This constraint is called at_least in MiniZinc.

Example

To ensure that 3 appears at least once in each of the subsets {a, b} and {b, c}:

julia> model = MOI.Utilities.Model{Float64}();

julia> a, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))

julia> b, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))

julia> c, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(3), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(3))

julia> x, d, set = [a, b, b, c], [2, 2], [3]
(MathOptInterface.VariableIndex[MOI.VariableIndex(1), MOI.VariableIndex(2), MOI.VariableIndex(2), MOI.VariableIndex(3)], [2, 2], [3])

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.CountAtLeast(1, d, Set(set)))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountAtLeast}(1)

CountBelongs(dimenson::Int, set::Set{Int})

The set $\{(n, x) \in \mathbb{Z}^{1+d}\}$, such that n elements of the vector x take on of the values in set and dimension = 1 + d.

Also known as

This constraint is called among by MiniZinc.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> n, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))

julia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)
 MOI.VariableIndex(4)

julia> set = Set([3, 4, 5])
Set{Int64} with 3 elements:
  5
  4
  3

julia> MOI.add_constraint(model, MOI.VectorOfVariables([n; x]), MOI.CountBelongs(4, set))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountBelongs}(1)

CountDistinct(dimension::Int)

The set $\{(n, x) \in \mathbb{Z}^{1+d}\}$, such that the number of distinct values in x is n and dimension = 1 + d.

Also known as

This constraint is called nvalues in MiniZinc.

Example

To model:

• if n == 1, then x[1] == x[2] == x[3]
• if n == 2, then
  • x[1] == x[2] != x[3] or
  • x[1] != x[2] == x[3] or
  • x[1] == x[3] != x[2]
• if n == 3, then x[1] != x[2], x[2] != x[3] and x[3] != x[1]

julia> model = MOI.Utilities.Model{Float64}();

julia> n, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))

julia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)
 MOI.VariableIndex(4)

julia> MOI.add_constraint(model, MOI.VectorOfVariables(vcat(n, x)), MOI.CountDistinct(4))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountDistinct}(1)

Relationship to AllDifferent

When the first element is d, CountDistinct is equivalent to an AllDifferent constraint.

CountGreaterThan(dimension::Int)

The set $\{(c, y, x) \in \mathbb{Z}^{1+1+d}\}$, such that c is strictly greater than the number of occurences of y in x and dimension = 1 + 1 + d.

Also known as

This constraint is called count_gt in MiniZinc.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> c, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))

julia> y, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))

julia> x = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(3)
 MOI.VariableIndex(4)
 MOI.VariableIndex(5)

julia> MOI.add_constraint(model, MOI.VectorOfVariables([c; y; x]), MOI.CountGreaterThan(5))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.CountGreaterThan}(1)

Cumulative(dimension::Int)

The set $\{(s, d, r, b) \in \mathbb{Z}^{3n+1}\}$, representing the cumulative global constraint, where n == length(s) == length(r) == length(b) and dimension = 3n + 1.

Cumulative requires that a set of tasks given by start times $s$, durations $d$, and resource requirements $r$, never requires more than the global resource bound $b$ at any one time.

Also known as

This constraint is called cumulative in MiniZinc.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> s = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)

julia> d = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(4)
 MOI.VariableIndex(5)
 MOI.VariableIndex(6)

julia> r = [MOI.add_constrained_variable(model, MOI.Integer())[1] for _ in 1:3]
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(7)
 MOI.VariableIndex(8)
 MOI.VariableIndex(9)

julia> b, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(10), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(10))

julia> MOI.add_constraint(model, MOI.VectorOfVariables([s; d; r; b]), MOI.Cumulative(10))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Cumulative}(1)

Path(from::Vector{Int}, to::Vector{Int})

Given a graph comprised of a set of nodes 1..N and a set of arcs 1..E represented by an edge from node from[i] to node to[i], Path constrains the set $(s, t, ns, es) \in (1..N)\times(1..E)\times\{0,1\}^N\times\{0,1\}^E$, to form subgraph that is a path from node s to node t, where node n is in the path if ns[n] is 1, and edge e is in the path if es[e] is 1.

The path must be acyclic, and it must traverse all nodes n for which ns[n] is 1, and all edges e for which es[e] is 1.

Also known as

This constraint is called path in MiniZinc.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> N, E = 4, 5
(4, 5)

julia> from = [1, 1, 2, 2, 3]
5-element Vector{Int64}:
 1
 1
 2
 2
 3

julia> to = [2, 3, 3, 4, 4]
5-element Vector{Int64}:
 2
 3
 3
 4
 4

julia> s, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(1))

julia> t, _ = MOI.add_constrained_variable(model, MOI.Integer())
(MOI.VariableIndex(2), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.Integer}(2))

julia> ns = MOI.add_variables(model, N)
4-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(3)
 MOI.VariableIndex(4)
 MOI.VariableIndex(5)
 MOI.VariableIndex(6)

julia> MOI.add_constraint.(model, ns, MOI.ZeroOne())
4-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}}:
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(3)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(4)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(5)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(6)

julia> es = MOI.add_variables(model, E)
5-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(7)
 MOI.VariableIndex(8)
 MOI.VariableIndex(9)
 MOI.VariableIndex(10)
 MOI.VariableIndex(11)

julia> MOI.add_constraint.(model, es, MOI.ZeroOne())
5-element Vector{MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}}:
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(7)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(8)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(9)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(10)
 MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(11)

julia> MOI.add_constraint(model, MOI.VectorOfVariables([s; t; ns; es]), MOI.Path(from, to))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Path}(1)

Reified(set::AbstractSet)

The constraint $[z; f(x)] \in Reified(S)$ ensures that $f(x) \in S$ if and only if $z == 1$, where $z \in \{0, 1\}$.

julia> model = MOI.Utilities.UniversalFallback(MOI.Utilities.Model{Float64}());

julia> z, _ = MOI.add_constrained_variable(model, MOI.ZeroOne())
(MOI.VariableIndex(1), MathOptInterface.ConstraintIndex{MathOptInterface.VariableIndex, MathOptInterface.ZeroOne}(1))

julia> x = MOI.add_variable(model)
MOI.VariableIndex(2)

julia> MOI.add_constraint(
           model,
           MOI.Utilities.vectorize([z, 2.0 * x]),
           MOI.Reified(MOI.GreaterThan(1.0)),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorAffineFunction{Float64}, MathOptInterface.Reified{MathOptInterface.GreaterThan{Float64}}}(1)

Table(table::Matrix{T}) where {T}

The set $\{x \in \mathbb{R}^d\}$ where d = size(table, 2), such that x belongs to one row of table. That is, there exists some j in 1:size(table, 1), such that x[i] = table[j, i] for all i=1:size(table, 2).

Also known as

This constraint is called table in MiniZinc.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> x = MOI.add_variables(model, 3)
3-element Vector{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(1)
 MOI.VariableIndex(2)
 MOI.VariableIndex(3)

julia> table = Float64[1 1 0; 0 1 1; 1 0 1; 1 1 1]
4×3 Matrix{Float64}:
 1.0  1.0  0.0
 0.0  1.0  1.0
 1.0  0.0  1.0
 1.0  1.0  1.0

julia> MOI.add_constraint(model, MOI.VectorOfVariables(x), MOI.Table(table))
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.Table{Float64}}(1)

abstract type AbstractSymmetricMatrixSetTriangle <: AbstractVectorSet end

Abstract supertype for subsets of the (vectorized) cone of symmetric matrices, with side_dimension rows and columns. The entries of the upper-right triangular part of the matrix are given column by column (or equivalently, the entries of the lower-left triangular part are given row by row). A vectorized cone of dimension $n$ corresponds to a square matrix with side dimension $\sqrt{1/4 + 2 n} - 1/2$. (Because a $d \times d$ matrix has $d(d + 1) / 2$ elements in the upper or lower triangle.)

Example

The matrix

\[\begin{bmatrix} 1 & 2 & 4\\ 2 & 3 & 5\\ 4 & 5 & 6 \end{bmatrix}\]

has side_dimension 3 and vectorization $(1, 2, 3, 4, 5, 6)$.

Note

Two packed storage formats exist for symmetric matrices, the respective orders of the entries are:

• upper triangular column by column (or lower triangular row by row);
• lower triangular column by column (or upper triangular row by row).

The advantage of the first format is the mapping between the (i, j) matrix indices and the k index of the vectorized form. It is simpler and does not depend on the side dimension of the matrix. Indeed,

• the entry of matrix indices (i, j) has vectorized index k = div((j - 1) * j, 2) + i if $i \leq j$ and k = div((i - 1) * i, 2) + j if $j \leq i$;
• and the entry with vectorized index k has matrix indices i = div(1 + isqrt(8k - 7), 2) and j = k - div((i - 1) * i, 2) or j = div(1 + isqrt(8k - 7), 2) and i = k - div((j - 1) * j, 2).

Duality note

The scalar product for the symmetric matrix in its vectorized form is the sum of the pairwise product of the diagonal entries plus twice the sum of the pairwise product of the upper diagonal entries; see [p. 634, 1]. This has important consequence for duality.

Consider for example the following problem (PositiveSemidefiniteConeTriangle is a subtype of AbstractSymmetricMatrixSetTriangle)

\[\begin{align*} & \max_{x \in \mathbb{R}} & x \\ & \;\;\text{s.t.} & (1, -x, 1) & \in \text{PositiveSemidefiniteConeTriangle}(2). \end{align*}\]

The dual is the following problem

\[\begin{align*} & \min_{x \in \mathbb{R}^3} & y_1 + y_3 \\ & \;\;\text{s.t.} & 2y_2 & = 1\\ & & y & \in \text{PositiveSemidefiniteConeTriangle}(2). \end{align*}\]

Why do we use $2y_2$ in the dual constraint instead of $y_2$ ? The reason is that $2y_2$ is the scalar product between $y$ and the symmetric matrix whose vectorized form is $(0, 1, 0)$. Indeed, with our modified scalar products we have

\[\langle (0, 1, 0), (y_1, y_2, y_3) \rangle = \mathrm{trace} \begin{pmatrix} 0 & 1\\ 1 & 0 \end{pmatrix} \begin{pmatrix} y_1 & y_2\\ y_2 & y_3 \end{pmatrix} = 2y_2.\]

References

[1] Boyd, S. and Vandenberghe, L.. Convex optimization. Cambridge university press, 2004.

abstract type AbstractSymmetricMatrixSetSquare <: AbstractVectorSet end

Abstract supertype for subsets of the (vectorized) cone of symmetric matrices, with side_dimension rows and columns. The entries of the matrix are given column by column (or equivalently, row by row). The matrix is both constrained to be symmetric and to have its triangular_form belong to the corresponding set. That is, if the functions in entries $(i, j)$ and $(j, i)$ are different, then a constraint will be added to make sure that the entries are equal.

Example

PositiveSemidefiniteConeSquare is a subtype of AbstractSymmetricMatrixSetSquare and constraining the matrix

\[\begin{bmatrix} 1 & -y\\ -z & 0\\ \end{bmatrix}\]

to be symmetric positive semidefinite can be achieved by constraining the vector $(1, -z, -y, 0)$ (or $(1, -y, -z, 0)$) to belong to the PositiveSemidefiniteConeSquare(2). It both constrains $y = z$ and $(1, -y, 0)$ (or $(1, -z, 0)$) to be in PositiveSemidefiniteConeTriangle(2), since triangular_form(PositiveSemidefiniteConeSquare) is PositiveSemidefiniteConeTriangle.

side_dimension(
    set::Union{
        AbstractSymmetricMatrixSetTriangle,
        AbstractSymmetricMatrixSetSquare,
        HermitianPositiveSemidefiniteConeTriangle,
    },
)

Side dimension of the matrices in set.

Convention

By convention, the side dimension should be stored in the side_dimension field. If this is not the case for a subtype of AbstractSymmetricMatrixSetTriangle, or AbstractSymmetricMatrixSetSquare you must implement this method.

triangular_form(S::Type{<:AbstractSymmetricMatrixSetSquare})
triangular_form(set::AbstractSymmetricMatrixSetSquare)

Return the AbstractSymmetricMatrixSetTriangle corresponding to the vectorization of the upper triangular part of matrices in the AbstractSymmetricMatrixSetSquare set.

PositiveSemidefiniteConeTriangle(side_dimension::Int) <: AbstractSymmetricMatrixSetTriangle

The (vectorized) cone of symmetric positive semidefinite matrices, with non-negative side_dimension rows and columns.

See AbstractSymmetricMatrixSetTriangle for more details on the vectorized form.

PositiveSemidefiniteConeSquare(side_dimension::Int) <: AbstractSymmetricMatrixSetSquare

The cone of symmetric positive semidefinite matrices, with non-negative side length side_dimension.

See AbstractSymmetricMatrixSetSquare for more details on the vectorized form.

The entries of the matrix are given column by column (or equivalently, row by row).

The matrix is both constrained to be symmetric and to be positive semidefinite. That is, if the functions in entries $(i, j)$ and $(j, i)$ are different, then a constraint will be added to make sure that the entries are equal.

Example

Constraining the matrix

\[\begin{bmatrix} 1 & -y\\ -z & 0\\ \end{bmatrix}\]

to be symmetric positive semidefinite can be achieved by constraining the vector $(1, -z, -y, 0)$ (or $(1, -y, -z, 0)$) to belong to the PositiveSemidefiniteConeSquare(2).

It both constrains $y = z$ and $(1, -y, 0)$ (or $(1, -z, 0)$) to be in PositiveSemidefiniteConeTriangle(2).

HermitianPositiveSemidefiniteConeTriangle(side_dimension::Int) <: AbstractVectorSet

The (vectorized) cone of Hermitian positive semidefinite matrices, with non-negative side_dimension rows and columns.

Becaue the matrix is Hermitian, the diagonal elements are real, and the complex-valued lower triangular entries are obtained as the conjugate of corresponding upper triangular entries.

Vectorization format

The vectorized form starts with real part of the entries of the upper triangular part of the matrix, given column by column as explained in AbstractSymmetricMatrixSetSquare.

It is then followed by the imaginary part of the off-diagonal entries of the upper triangular part, also given column by column.

For example, the matrix

\[\begin{bmatrix} 1 & 2 + 7im & 4 + 8im\\ 2 - 7im & 3 & 5 + 9im\\ 4 - 8im & 5 - 9im & 6 \end{bmatrix}\]

has side_dimension 3 and is represented as the vector $[1, 2, 3, 4, 5, 6, 7, 8, 9]$.

LogDetConeTriangle(side_dimension::Int)

The log-determinant cone $\{ (t, u, X) \in \mathbb{R}^{2 + d(d+1)/2} : t \le u \log(\det(X/u)), u > 0 \}$, where the matrix X is represented in the same symmetric packed format as in the PositiveSemidefiniteConeTriangle.

The non-negative argument side_dimension is the side dimension of the matrix X, that is, its number of rows or columns.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> X = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; X]),
           MOI.LogDetConeTriangle(2),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.LogDetConeTriangle}(1)

LogDetConeSquare(side_dimension::Int)

The log-determinant cone $\{ (t, u, X) \in \mathbb{R}^{2 + d^2} : t \le u \log(\det(X/u)), X \text{ symmetric}, u > 0 \}$, where the matrix X is represented in the same format as in the PositiveSemidefiniteConeSquare.

Similarly to PositiveSemidefiniteConeSquare, constraints are added to ensure that X is symmetric.

The non-negative argument side_dimension is the side dimension of the matrix X, that is, its number of rows or columns.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> X = reshape(MOI.add_variables(model, 4), 2, 2)
2×2 Matrix{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(2)  MOI.VariableIndex(4)
 MOI.VariableIndex(3)  MOI.VariableIndex(5)

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; vec(X)]),
           MOI.LogDetConeSquare(2),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.LogDetConeSquare}(1)

RootDetConeTriangle(side_dimension::Int)

The root-determinant cone $\{ (t, X) \in \mathbb{R}^{1 + d(d+1)/2} : t \le \det(X)^{1/d} \}$, where the matrix X is represented in the same symmetric packed format as in the PositiveSemidefiniteConeTriangle.

The non-negative argument side_dimension is the side dimension of the matrix X, that is, its number of rows or columns.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> X = MOI.add_variables(model, 3);

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; X]),
           MOI.RootDetConeTriangle(2),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RootDetConeTriangle}(1)

RootDetConeSquare(side_dimension::Int)

The root-determinant cone $\{ (t, X) \in \mathbb{R}^{1 + d^2} : t \le \det(X)^{1/d}, X \text{ symmetric} \}$, where the matrix X is represented in the same format as PositiveSemidefiniteConeSquare.

Similarly to PositiveSemidefiniteConeSquare, constraints are added to ensure that X is symmetric.

The non-negative argument side_dimension is the side dimension of the matrix X, that is, its number of rows or columns.

Example

julia> model = MOI.Utilities.Model{Float64}();

julia> t = MOI.add_variable(model)
MOI.VariableIndex(1)

julia> X = reshape(MOI.add_variables(model, 4), 2, 2)
2×2 Matrix{MathOptInterface.VariableIndex}:
 MOI.VariableIndex(2)  MOI.VariableIndex(4)
 MOI.VariableIndex(3)  MOI.VariableIndex(5)

julia> MOI.add_constraint(
           model,
           MOI.VectorOfVariables([t; vec(X)]),
           MOI.RootDetConeSquare(2),
       )
MathOptInterface.ConstraintIndex{MathOptInterface.VectorOfVariables, MathOptInterface.RootDetConeSquare}(1)