Constraints

@constraint(m::Model, expr, kw_args...)

Add a constraint described by the expression expr.

@constraint(m::Model, ref[i=..., j=..., ...], expr, kw_args...)

Add a group of constraints described by the expression expr parametrized by i, j, ...

The expression expr can either be

• of the form func in set constraining the function func to belong to the set set which is either a MOI.AbstractSet or one of the JuMP shortcuts SecondOrderCone, RotatedSecondOrderCone and PSDCone, e.g. @constraint(model, [1, x-1, y-2] in SecondOrderCone()) constrains the norm of [x-1, y-2] be less than 1;
• of the form a sign b, where sign is one of ==, ≥, >=, ≤ and <= building the single constraint enforcing the comparison to hold for the expression a and b, e.g. @constraint(m, x^2 + y^2 == 1) constrains x and y to lie on the unit circle;
• of the form a ≤ b ≤ c or a ≥ b ≥ c (where ≤ and <= (resp. ≥ and >=) can be used interchangeably) constraining the paired the expression b to lie between a and c;
• of the forms @constraint(m, a .sign b) or @constraint(m, a .sign b .sign c) which broadcast the constraint creation to each element of the vectors.

The recognized keyword arguments in kw_args are the following:

• base_name: Sets the name prefix used to generate constraint names. It corresponds to the constraint name for scalar constraints, otherwise, the constraint names are set to base_name[...] for each index ... of the axes axes.
• container: Specify the container type.

Note for extending the constraint macro

Each constraint will be created using add_constraint(m, build_constraint(_error, func, set)) where

• _error is an error function showing the constraint call in addition to the error message given as argument,
• func is the expression that is constrained
• and set is the set in which it is constrained to belong.

For expr of the first type (i.e. @constraint(m, func in set)), func and set are passed unchanged to build_constraint but for the other types, they are determined from the expressions and signs. For instance, @constraint(m, x^2 + y^2 == 1) is transformed into add_constraint(m, build_constraint(_error, x^2 + y^2, MOI.EqualTo(1.0))).

To extend JuMP to accept new constraints of this form, it is necessary to add the corresponding methods to build_constraint. Note that this will likely mean that either func or set will be some custom type, rather than e.g. a Symbol, since we will likely want to dispatch on the type of the function or set appearing in the constraint.

For extensions that need to create constraints with more information than just func and set, an additional positional argument can be specified to @constraint that will then be passed on build_constraint. Hence, we can enable this syntax by defining extensions of build_constraint(_error, func, set, my_arg; kw_args...). This produces the user syntax: @constraint(model, ref[...], expr, my_arg, kw_args...).

@constraints(model, args...)

Adds groups of constraints at once, in the same fashion as the @constraint macro.

The model must be the first argument, and multiple constraints can be added on multiple lines wrapped in a begin ... end block.

Examples

@constraints(model, begin
    x >= 1
    y - w <= 2
    sum_to_one[i=1:3], z[i] + y == 1
end)

@SDconstraint(model::Model, expr)

Add a semidefinite constraint described by the expression expr.

@SDconstraint(model::Model, ref[i=..., j=..., ...], expr)

Add a group of semidefinite constraints described by the expression expr parametrized by i, j, ...

The expression expr needs to be of the form a sign b where sign is ⪰, ≥, >=, ⪯, ≤ or <= and a and b are square matrices. It constrains the square matrix x (or -x if the sign is ⪯, ≤ or <=) to be symmetric and positive semidefinite where

• x = a, if b is the symbol 0,
• x = -b, if a is the symbol 0,
• otherwise, x = a - b.

By default, we check numerical symmetry of the matrix x, and if symmetry is violated by some arbitrary amount, we add explicit equality constraints. You can use Symmetric(x) in PSDCone() with the @constraint macro to skip these checks if you know the matrix must be symmetric; see PSDCone for more information.

Examples

The following constrains the matrix [x-1 2x-2; -3 x-4] to be symmetric and positive semidefinite, that is, it constrains 2x-2 to be equal to -3 and constrains all eigenvalues of the matrix to be nonnegative.

julia> model = Model();

julia> @variable(model, x)
x

julia> a = [x 2x
            0  x];

julia> b = [1 2
            3 4];

julia> cref = @SDconstraint(model, a ⪰ b)
[x - 1  2 x - 2;
 -3     x - 4  ] ∈ PSDCone()

julia> jump_function(constraint_object(cref))
4-element Array{GenericAffExpr{Float64,VariableRef},1}:
 x - 1
 -3
 2 x - 2
 x - 4

julia> moi_set(constraint_object(cref))
MathOptInterface.PositiveSemidefiniteConeSquare(2)

In the set PositiveSemidefiniteConeSquare(2) in the last output, Square means that the matrix is passed as a square matrix as the corresponding off-diagonal entries need to be constrained to be equal. A similar set PositiveSemidefiniteConeTriangle exists which only uses the upper triangular part of the matrix assuming that it is symmetric, see PSDCone to see how to use it.

@SDconstraints(model, args...)

Adds multiple semi-definite constraints to model at once, in the same fashion as the @SDconstraint macro.

The model must be the first argument, and multiple constraints can be added on multiple lines wrapped in a begin ... end block.

Examples

@SDconstraints(model, begin
    A * x >= b
    b - C * y >= 0
end)

ConstraintRef

Holds a reference to the model and the corresponding MOI.ConstraintIndex.

abstract type AbstractConstraint

An abstract base type for all constraint types. AbstractConstraints store the function and set directly, unlike ConstraintRefs that are merely references to constraints stored in a model. AbstractConstraints do not need to be attached to a model.

struct ScalarConstraint

The data for a scalar constraint. The func field contains a JuMP object representing the function and the set field contains the MOI set. See also the documentation on JuMP's representation of constraints for more background.

struct VectorConstraint

The data for a vector constraint. The func field contains a JuMP object representing the function and the set field contains the MOI set. The shape field contains an AbstractShape matching the form in which the constraint was constructed (e.g., by using matrices or flat vectors). See also the documentation on JuMP's representation of constraints.

name(con_ref::ConstraintRef)

Get a constraint's name attribute.

set_name(con_ref::ConstraintRef, s::AbstractString)

Set a constraint's name attribute.

constraint_by_name(model::AbstractModel,
                   name::String)::Union{ConstraintRef, Nothing}

Return the reference of the constraint with name attribute name or Nothing if no constraint has this name attribute. Throws an error if several constraints have name as their name attribute.

constraint_by_name(model::AbstractModel,
                   name::String,
                   F::Type{<:Union{AbstractJuMPScalar,
                                   Vector{<:AbstractJuMPScalar},
                                   MOI.AbstactFunction}},
                   S::Type{<:MOI.AbstractSet})::Union{ConstraintRef, Nothing}

Similar to the method above, except that it throws an error if the constraint is not an F-in-S contraint where F is either the JuMP or MOI type of the function, and S is the MOI type of the set. This method is recommended if you know the type of the function and set since its returned type can be inferred while for the method above (i.e. without F and S), the exact return type of the constraint index cannot be inferred.

julia> using JuMP

julia> model = Model()
A JuMP Model
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: NO_OPTIMIZER
Solver name: No optimizer attached.

julia> @variable(model, x)
x

julia> @constraint(model, con, x^2 == 1)
con : x² = 1.0

julia> constraint_by_name(model, "kon")

julia> constraint_by_name(model, "con")
con : x² = 1.0

julia> constraint_by_name(model, "con", AffExpr, MOI.EqualTo{Float64})

julia> constraint_by_name(model, "con", QuadExpr, MOI.EqualTo{Float64})
con : x² = 1.0

normalized_coefficient(con_ref::ConstraintRef, variable::VariableRef)

Return the coefficient associated with variable in constraint after JuMP has normalized the constraint into its standard form. See also set_normalized_coefficient.

set_normalized_coefficient(con_ref::ConstraintRef, variable::VariableRef, value)

Set the coefficient of variable in the constraint constraint to value.

Note that prior to this step, JuMP will aggregate multiple terms containing the same variable. For example, given a constraint 2x + 3x <= 2, set_normalized_coefficient(con, x, 4) will create the constraint 4x <= 2.

model = Model()
@variable(model, x)
@constraint(model, con, 2x + 3x <= 2)
set_normalized_coefficient(con, x, 4)
con

# output

con : 4 x <= 2.0

normalized_rhs(con_ref::ConstraintRef)

Return the right-hand side term of con_ref after JuMP has converted the constraint into its normalized form. See also set_normalized_rhs.

set_normalized_rhs(con_ref::ConstraintRef, value)

Set the right-hand side term of constraint to value.

Note that prior to this step, JuMP will aggregate all constant terms onto the right-hand side of the constraint. For example, given a constraint 2x + 1 <= 2, set_normalized_rhs(con, 4) will create the constraint 2x <= 4, not 2x + 1 <= 4.

julia> @constraint(model, con, 2x + 1 <= 2)
con : 2 x <= 1.0

julia> set_normalized_rhs(con, 4)

julia> con
con : 2 x <= 4.0

add_to_function_constant(constraint::ConstraintRef, value)

Add value to the function constant term.

Note that for scalar constraints, JuMP will aggregate all constant terms onto the right-hand side of the constraint so instead of modifying the function, the set will be translated by -value. For example, given a constraint 2x <= 3, add_to_function_constant(c, 4) will modify it to 2x <= -1.

Examples

For scalar constraints, the set is translated by -value:

julia> @constraint(model, con, 0 <= 2x - 1 <= 2)
con : 2 x ∈ [1.0, 3.0]

julia> add_to_function_constant(con, 4)

julia> con
con : 2 x ∈ [-3.0, -1.0]

For vector constraints, the constant is added to the function:

julia> @constraint(model, con, [x + y, x, y] in SecondOrderCone())
con : [x + y, x, y] ∈ MathOptInterface.SecondOrderCone(3)

julia> add_to_function_constant(con, [1, 2, 2])

julia> con
con : [x + y + 1, x + 2, y + 2] ∈ MathOptInterface.SecondOrderCone(3)

delete(model::Model, con_ref::ConstraintRef)

Delete the constraint associated with constraint_ref from the model model.

Note that delete does not unregister the name from the model, so adding a new constraint of the same name will throw an error. Use unregister to unregister the name after deletion as follows:

@constraint(model, c, 2x <= 1)
delete(model, c)
unregister(model, :c)

See also: unregister

delete(model::Model, con_refs::Vector{<:ConstraintRef})

Delete the constraints associated with con_refs from the model model. Solvers may implement specialized methods for deleting multiple constraints of the same concrete type, i.e., when isconcretetype(eltype(con_refs)). These may be more efficient than repeatedly calling the single constraint delete method.

See also: unregister

delete(model::Model, variable_ref::VariableRef)

Delete the variable associated with variable_ref from the model model.

Note that delete does not unregister the name from the model, so adding a new variable of the same name will throw an error. Use unregister to unregister the name after deletion as follows:

@variable(model, x)
delete(model, x)
unregister(model, :x)

See also: unregister

delete(model::Model, variable_refs::Vector{VariableRef})

Delete the variables associated with variable_refs from the model model. Solvers may implement methods for deleting multiple variables that are more efficient than repeatedly calling the single variable delete method.

See also: unregister

is_valid(model::Model, con_ref::ConstraintRef{<:AbstractModel})

Return true if constraint_ref refers to a valid constraint in model.

is_valid(model::Model, variable_ref::VariableRef)

Return true if variable refers to a valid variable in model.

is_valid(model::Model, c::ConstraintRef{Model,NonlinearConstraintIndex})

Return true if c refers to a valid nonlinear constraint in model.

struct ConstraintNotOwned{C <: ConstraintRef} <: Exception
    constraint_ref::C
end

The constraint constraint_ref was used in a model different to owner_model(constraint_ref).

list_of_constraint_types(model::Model)::Vector{Tuple{DataType, DataType}}

Return a list of tuples of the form (F, S) where F is a JuMP function type and S is an MOI set type such that all_constraints(model, F, S) returns a nonempty list.

Example

julia> model = Model();

julia> @variable(model, x >= 0, Bin);

julia> @constraint(model, 2x <= 1);

julia> list_of_constraint_types(model)
3-element Array{Tuple{DataType,DataType},1}:
 (GenericAffExpr{Float64,VariableRef}, MathOptInterface.LessThan{Float64})
 (VariableRef, MathOptInterface.GreaterThan{Float64})
 (VariableRef, MathOptInterface.ZeroOne)

all_constraints(model::Model, function_type, set_type)::Vector{<:ConstraintRef}

Return a list of all constraints currently in the model where the function has type function_type and the set has type set_type. The constraints are ordered by creation time.

See also list_of_constraint_types and num_constraints.

Example

julia> model = Model();

julia> @variable(model, x >= 0, Bin);

julia> @constraint(model, 2x <= 1);

julia> all_constraints(model, VariableRef, MOI.GreaterThan{Float64})
1-element Array{ConstraintRef{Model,MathOptInterface.ConstraintIndex{MathOptInterface.SingleVariable,MathOptInterface.GreaterThan{Float64}},ScalarShape},1}:
 x ≥ 0.0

julia> all_constraints(model, VariableRef, MOI.ZeroOne)
1-element Array{ConstraintRef{Model,MathOptInterface.ConstraintIndex{MathOptInterface.SingleVariable,MathOptInterface.ZeroOne},ScalarShape},1}:
 x binary

julia> all_constraints(model, AffExpr, MOI.LessThan{Float64})
1-element Array{ConstraintRef{Model,MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64},MathOptInterface.LessThan{Float64}},ScalarShape},1}:
 2 x ≤ 1.0

num_constraints(model::Model, function_type, set_type)::Int64

Return the number of constraints currently in the model where the function has type function_type and the set has type set_type.

See also list_of_constraint_types and all_constraints.

Example

julia> model = Model();

julia> @variable(model, x >= 0, Bin);

julia> @variable(model, y);

julia> @constraint(model, y in MOI.GreaterThan(1.0));

julia> @constraint(model, y <= 1.0);

julia> @constraint(model, 2x <= 1);

julia> num_constraints(model, VariableRef, MOI.GreaterThan{Float64})
2

julia> num_constraints(model, VariableRef, MOI.ZeroOne)
1

julia> num_constraints(model, AffExpr, MOI.LessThan{Float64})
2

index(cr::ConstraintRef)::MOI.ConstraintIndex

Return the index of the constraint that corresponds to cr in the MOI backend.

optimizer_index(cr::ConstraintRef{Model})::MOI.ConstraintIndex

Return the index of the constraint that corresponds to cr in the optimizer model. It throws NoOptimizer if no optimizer is set and throws an ErrorException if the optimizer is set but is not attached or if the constraint is bridged.

constraint_object(con_ref::ConstraintRef)

Return the underlying constraint data for the constraint referenced by ref.

set_dual_start_value(con_ref::ConstraintRef, value)

Set the dual start value (MOI attribute ConstraintDualStart) of the constraint con_ref to value. To remove a dual start value set it to nothing.

See also dual_start_value.

dual_start_value(con_ref::ConstraintRef)

Return the dual start value (MOI attribute ConstraintDualStart) of the constraint con_ref.

Note: If no dual start value has been set, dual_start_value will return nothing.

See also set_dual_start_value.

SecondOrderCone

Second order cone object that can be used to constrain the euclidean norm of a vector x to be less than or equal to a nonnegative scalar t. This is a shortcut for the MOI.SecondOrderCone.

Examples

The following constrains $\|(x-1, x-2)\|_2 \le t$ and $t \ge 0$:

julia> model = Model();

julia> @variable(model, x)
x

julia> @variable(model, t)
t

julia> @constraint(model, [t, x-1, x-2] in SecondOrderCone())
[t, x - 1, x - 2] ∈ MathOptInterface.SecondOrderCone(3)

RotatedSecondOrderCone

Rotated second order cone object that can be used to constrain the square of the euclidean norm of a vector x to be less than or equal to $2tu$ where t and u are nonnegative scalars. This is a shortcut for the MOI.RotatedSecondOrderCone.

Examples

The following constrains $\|(x-1, x-2)\|_2 \le 2tx$ and $t, x \ge 0$:

julia> model = Model();

julia> @variable(model, x)
x

julia> @variable(model, t)
t

julia> @constraint(model, [t, x, x-1, x-2] in RotatedSecondOrderCone())
[t, x, x - 1, x - 2] ∈ MathOptInterface.RotatedSecondOrderCone(4)

PSDCone

Positive semidefinite cone object that can be used to constrain a square matrix to be positive semidefinite in the @constraint macro. If the matrix has type Symmetric then the columns vectorization (the vector obtained by concatenating the columns) of its upper triangular part is constrained to belong to the MOI.PositiveSemidefiniteConeTriangle set, otherwise its column vectorization is constrained to belong to the MOI.PositiveSemidefiniteConeSquare set.

Examples

Consider the following example:

julia> model = Model();

julia> @variable(model, x)
x

julia> a = [ x 2x
            2x  x];

julia> b = [1 2
            2 4];

julia> cref = @SDconstraint(model, a ⪰ b)
[x - 1    2 x - 2;
 2 x - 2  x - 4  ] ∈ PSDCone()

julia> jump_function(constraint_object(cref))
4-element Array{GenericAffExpr{Float64,VariableRef},1}:
 x - 1
 2 x - 2
 2 x - 2
 x - 4

julia> moi_set(constraint_object(cref))
MathOptInterface.PositiveSemidefiniteConeSquare(2)

We see in the output of the last command that the matrix the vectorization of the matrix is constrained to belong to the PositiveSemidefiniteConeSquare.

julia> using LinearAlgebra # For Symmetric

julia> cref = @constraint(model, Symmetric(a - b) in PSDCone())
[x - 1    2 x - 2;
 2 x - 2  x - 4  ] ∈ PSDCone()

julia> jump_function(constraint_object(cref))
3-element Array{GenericAffExpr{Float64,VariableRef},1}:
 x - 1
 2 x - 2
 x - 4

julia> moi_set(constraint_object(cref))
MathOptInterface.PositiveSemidefiniteConeTriangle(2)

As we see in the output of the last command, the vectorization of only the upper triangular part of the matrix is constrained to belong to the PositiveSemidefiniteConeSquare.

SOS1

SOS1 (Special Ordered Sets type 1) object than can be used to constrain a vector x to a set where at most 1 variable can take a non-zero value, all others being at 0. The weights, when specified, induce an ordering of the variables; as such, they should be unique values. The kth element in the set corresponds to the kth weight in weights. See here for a description of SOS constraints and their potential uses. This is a shortcut for the MathOptInterface.SOS1 set.

SOS2

SOS1 (Special Ordered Sets type 2) object than can be used to constrain a vector x to a set where at most 2 variables can take a non-zero value, all others being at 0. In addition, if two are non-zero these must be consecutive in their ordering. The weights induce an ordering of the variables; as such, they should be unique values. The kth element in the set corresponds to the kth weight in weights. See here for a description of SOS constraints and their potential uses. This is a shortcut for the MathOptInterface.SOS2 set.

SkewSymmetricMatrixSpace()

Use in the @variable macro to constrain a matrix of variables to be skew-symmetric.

Examples

@variable(model, Q[1:2, 1:2] in SkewSymmetricMatrixSpace())

SkewSymmetricMatrixShape

Shape object for a skew symmetric square matrix of side_dimension rows and columns. The vectorized form contains the entries of the upper-right triangular part of the matrix (without the diagonal) given column by column (or equivalently, the entries of the lower-left triangular part given row by row). The diagonal is zero.

moi_set(constraint::AbstractConstraint)

Return the set of the constraint constraint in the function-in-set form as a MathOptInterface.AbstractSet.

moi_set(s::AbstractVectorSet, dim::Int)

Returns the MOI set of dimension dim corresponding to the JuMP set s.

function_string(print_mode::Type{<:JuMP.PrintMode},
                func::Union{JuMP.AbstractJuMPScalar,
                            Vector{<:JuMP.AbstractJuMPScalar}})

Return a String representing the function func using print mode print_mode.

constraints_string(print_mode, model::AbstractModel)::Vector{String}

Return a list of Strings describing each constraint of the model.

in_set_string(print_mode::Type{<:PrintMode}, set)

Return a String representing the membership to the set set using print mode print_mode.

show_constraints_summary(io::IO, model::AbstractModel)

Write to io a summary of the number of constraints.