Variables

What is a JuMP variable?

The term variable in mathematical optimization has many meanings. Here, we distinguish between the following three types of variables:

1. optimization variables, which are the mathematical $x$ in the problem $\max\{f_0(x) | f_i(x) \in S_i\}$.
2. Julia variables, which are bindings between a name and a value, for example x = 1. (See here for the Julia docs.)
3. JuMP variables, which are instances of the VariableRef struct defined by JuMP that contains a reference to an optimization variable in a model. (Extra for experts: the VariableRef struct is a thin wrapper around a MOI.VariableIndex, and also contains a reference to the JuMP model.)

To illustrate these three types of variables, consider the following JuMP code (the full syntax is explained below):

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[1:2])
2-element Array{VariableRef,1}:
 x[1]
 x[2]

This code does three things:

1. It adds two optimization variables to model.
2. It creates two JuMP variables that act as references to those optimization variables.
3. It binds those JuMP variables as a vector with two elements to the Julia variable x.

To reduce confusion, we will attempt, where possible, to always refer to variables with their corresponding prefix.

JuMP variables can have attributes, such as names or an initial primal start value. We illustrate the name attribute in the following example:

julia> @variable(model, y, base_name="decision variable")
decision variable

This code does four things:

1. It adds one optimization variable to model.
2. It creates one JuMP variable that acts as a reference to that optimization variable.
3. It binds the JuMP variable to the Julia variable y.
4. It tells JuMP that the name attribute of this JuMP variable is "decision variable". JuMP uses the value of base_name when it has to print the variable as a string.

For example, when we print y at the REPL we get:

julia> y
decision variable

Because y is a Julia variable, we can bind it to a different value. For example, if we write:

julia> y = 1
1

y is no longer a binding to a JuMP variable. This does not mean that the JuMP variable has been destroyed. It still exists and is still a reference to the same optimization variable. The binding can be reset by querying the model for the symbol as it was written in the @variable macro. For example:

julia> model[:y]
decision variable

This act of looking up the JuMP variable by using the symbol is most useful when composing JuMP models across multiple functions, as illustrated by the following example:

function add_component_to_model(model::JuMP.Model)
    x = model[:x]
    # ... code that uses x
end
function build_model()
    model = Model()
    @variable(model, x)
    add_component_to_model(model)
end

Now that we understand the difference between optimization, JuMP, and Julia variables, we can introduce more of the functionality of the @variable macro.

Variable bounds

We have already seen the basic usage of the @variable macro. The next extension is to add lower- and upper-bounds to each optimization variable. This can be done as follows:

julia> @variable(model, x_free)
x_free

julia> @variable(model, x_lower >= 0)
x_lower

julia> @variable(model, x_upper <= 1)
x_upper

julia> @variable(model, 2 <= x_interval <= 3)
x_interval

julia> @variable(model, x_fixed == 4)
x_fixed

In the above examples, x_free represents an unbounded optimization variable, x_lower represents an optimization variable with a lower bound and so forth.

@variable(model, 1 <= x)  # works
a = 1
@variable(model, a <= x)  # errors

We can query whether an optimization variable has a lower- or upper-bound via the has_lower_bound and has_upper_bound functions. For example:

julia> has_lower_bound(x_free)
false

julia> has_upper_bound(x_upper)
true

If a variable has a lower or upper bound, we can query the value of it via the lower_bound and upper_bound functions. For example:

julia> lower_bound(x_interval)
2.0

julia> upper_bound(x_interval)
3.0

Querying the value of a bound that does not exist will result in an error.

Instead of using the <= and >= syntax, we can also use the lower_bound and upper_bound keyword arguments. For example:

julia> @variable(model, x, lower_bound=1, upper_bound=2)
x

julia> lower_bound(x)
1.0

Another option is to use the set_lower_bound and set_upper_bound functions. These can also be used to modify an existing variable bound. For example:

julia> @variable(model, x >= 1)
x

julia> lower_bound(x)
1.0

julia> set_lower_bound(x, 2)

julia> lower_bound(x)
2.0

We can delete variable bounds using delete_lower_bound and delete_upper_bound:

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

julia> lower_bound(x)
1.0

julia> delete_lower_bound(x)

julia> has_lower_bound(x)
false

julia> upper_bound(x)
2.0

julia> delete_upper_bound(x)

julia> has_upper_bound(x)
false

In addition to upper and lower bounds, JuMP variables can also be fixed to a value using fix. See also is_fixed, fix_value, and unfix.

julia> @variable(model, x == 1)
x

julia> is_fixed(x)
true

julia> fix_value(x)
1.0

julia> unfix(x)

julia> is_fixed(x)
false

Fixing a variable with existing bounds will throw an error. To delete the bounds prior to fixing, use fix(variable, value; force = true).

julia> @variable(model, x >= 1)
x

julia> fix(x, 2)
ERROR: Unable to fix x to 2 because it has existing variable bounds. Consider calling `JuMP.fix(variable, value; force=true)` which will delete existing bounds before fixing the variable.

julia> fix(x, 2; force = true)


julia> fix_value(x)
2.0

Variable names

The name, i.e. the value of the MOI.VariableName attribute, of a variable can be obtained by JuMP.name(::JuMP.VariableRef) and set by JuMP.set_name(::JuMP.VariableRef, ::String).

name(v::VariableRef)::String

set_name(v::VariableRef, s::AbstractString)

The variable can also be retrieved from its name using JuMP.variable_by_name.

variable_by_name(model::AbstractModel,
                 name::String)::Union{AbstractVariableRef, Nothing}

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> variable_by_name(model, "x")
x

julia> @variable(model, base_name="x")
x

julia> variable_by_name(model, "x")
ERROR: Multiple variables have the name x.
Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] get(::JuMP._MOIModel{Float64}, ::Type{MathOptInterface.VariableIndex}, ::String) at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/model.jl:222
 [3] get at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/universalfallback.jl:201 [inlined]
 [4] get(::MathOptInterface.Utilities.CachingOptimizer{MathOptInterface.AbstractOptimizer,MathOptInterface.Utilities.UniversalFallback{JuMP._MOIModel{Float64}}}, ::Type{MathOptInterface.VariableIndex}, ::String) at /home/blegat/.julia/dev/MathOptInterface/src/Utilities/cachingoptimizer.jl:490
 [5] variable_by_name(::Model, ::String) at /home/blegat/.julia/dev/JuMP/src/variables.jl:268
 [6] top-level scope at none:0

julia> var = @variable(model, base_name="y")
y

julia> variable_by_name(model, "y")
y

julia> set_name(var, "z")

julia> variable_by_name(model, "y")

julia> variable_by_name(model, "z")
z

julia> @variable(model, u[1:2])
2-element Array{VariableRef,1}:
 u[1]
 u[2]

julia> variable_by_name(model, "u[2]")
u[2]

Variable containers

In the examples above, we have mostly created scalar variables. By scalar, we mean that the Julia variable is bound to exactly one JuMP variable. However, it is often useful to create collections of JuMP variables inside more complicated data structures.

JuMP provides a mechanism for creating three types of these data structures, which we refer to as containers. The three types are Arrays, DenseAxisArrays, and SparseAxisArrays. We explain each of these in the following.

Arrays

We have already seen the creation of an array of JuMP variables with the x[1:2] syntax. This can naturally be extended to create multi-dimensional arrays of JuMP variables. For example:

julia> @variable(model, x[1:2, 1:2])
2×2 Array{VariableRef,2}:
 x[1,1]  x[1,2]
 x[2,1]  x[2,2]

Arrays of JuMP variables can be indexed and sliced as follows:

julia> x[1, 2]
x[1,2]

julia> x[2, :]
2-element Array{VariableRef,1}:
 x[2,1]
 x[2,2]

Variable bounds can depend upon the indices:

julia> @variable(model, x[i=1:2, j=1:2] >= 2i + j)
2×2 Array{VariableRef,2}:
 x[1,1]  x[1,2]
 x[2,1]  x[2,2]

julia> lower_bound.(x)
2×2 Array{Float64,2}:
 3.0  4.0
 5.0  6.0

JuMP will form an Array of JuMP variables when it can determine at compile time that the indices are one-based integer ranges. Therefore x[1:b] will create an Array of JuMP variables, but x[a:b] will not. If JuMP cannot determine that the indices are one-based integer ranges (e.g., in the case of x[a:b]), JuMP will create a DenseAxisArray instead.

DenseAxisArrays

We often want to create arrays where the indices are not one-based integer ranges. For example, we may want to create a variable indexed by the name of a product or a location. The syntax is the same as that above, except with an arbitrary vector as an index as opposed to a one-based range. The biggest difference is that instead of returning an Array of JuMP variables, JuMP will return a DenseAxisArray. For example:

julia> @variable(model, x[1:2, [:A,:B]])
2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
    Dimension 1, 1:2
    Dimension 2, Symbol[:A, :B]
And data, a 2×2 Array{VariableRef,2}:
 x[1,A]  x[1,B]
 x[2,A]  x[2,B]

DenseAxisArrays can be indexed and sliced as follows:

julia> x[1, :A]
x[1,A]

julia> x[2, :]
1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
    Dimension 1, Symbol[:A, :B]
And data, a 2-element Array{VariableRef,1}:
 x[2,A]
 x[2,B]

Similarly to the Array case, bounds can depend upon indices. For example:

julia> @variable(model, x[i=2:3, j=1:2:3] >= 0.5i + j)
2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
    Dimension 1, 2:3
    Dimension 2, 1:2:3
And data, a 2×2 Array{VariableRef,2}:
 x[2,1]  x[2,3]
 x[3,1]  x[3,3]

julia> lower_bound.(x)
2-dimensional DenseAxisArray{Float64,2,...} with index sets:
    Dimension 1, 2:3
    Dimension 2, 1:2:3
And data, a 2×2 Array{Float64,2}:
 2.0  4.0
 2.5  4.5

SparseAxisArrays

The third container type that JuMP natively supports is SparseAxisArray. These arrays are created when the indices do not form a rectangular set. For example, this applies when indices have a dependence upon previous indices (called triangular indexing). JuMP supports this as follows:

julia> @variable(model, x[i=1:2, j=i:2])
JuMP.Containers.SparseAxisArray{VariableRef,2,Tuple{Any,Any}} with 3 entries:
  [1, 2]  =  x[1,2]
  [2, 2]  =  x[2,2]
  [1, 1]  =  x[1,1]

We can also conditionally create variables via a JuMP-specific syntax. This syntax appends a comparison check that depends upon the named indices and is separated from the indices by a semi-colon (;). For example:

julia> @variable(model, x[i=1:4; mod(i, 2)==0])
JuMP.Containers.SparseAxisArray{VariableRef,1,Tuple{Any}} with 2 entries:
  [4]  =  x[4]
  [2]  =  x[2]

Forcing the container type

When creating a container of JuMP variables, JuMP will attempt to choose the tightest container type that can store the JuMP variables. Thus, it will prefer to create an Array before a DenseAxisArray and a DenseAxisArray before a SparseAxisArray. However, because this happens at compile time, it does not always make the best choice. To illustrate this, consider the following example:

julia> A = 1:2
1:2

julia> @variable(model, x[A])
1-dimensional DenseAxisArray{VariableRef,1,...} with index sets:
    Dimension 1, 1:2
And data, a 2-element Array{VariableRef,1}:
 x[1]
 x[2]

Since the value (and type) of A is unknown at parsing time, JuMP is unable to infer that A is a one-based integer range. Therefore, JuMP creates a DenseAxisArray, even though it could store these two variables in a standard one-dimensional Array.

We can share our knowledge that it is possible to store these JuMP variables as an array by setting the container keyword:

julia> @variable(model, y[A], container=Array)
2-element Array{VariableRef,1}:
 y[1]
 y[2]

JuMP now creates a vector of JuMP variables instead of a DenseAxisArray. Note that choosing an invalid container type will throw an error.

Integrality shortcuts

Adding integrality constraints to a model such as @constraint(model, x in MOI.ZeroOne()) and @constraint(model, x in MOI.Integer()) is a common operation. Therefore, JuMP supports two shortcuts for adding such constraints.

Binary (ZeroOne) constraints

Binary optimization variables are constrained to the set $x \in \{0, 1\}$. (The MOI.ZeroOne set in MathOptInterface.) Binary optimization variables can be created in JuMP by passing Bin as an optional positional argument:

julia> @variable(model, x, Bin)
x

We can check if an optimization variable is binary by calling is_binary on the JuMP variable, and binary constraints can be removed with unset_binary.

julia> is_binary(x)
true

julia> unset_binary(x)

julia> is_binary(x)
false

Binary optimization variables can also be created by setting the binary keyword to true.

julia> @variable(model, x, binary=true)
x

Integer constraints

Integer optimization variables are constrained to the set $x \in \mathbb{Z}$. (The MOI.Integer set in MathOptInterface.) Integer optimization variables can be created in JuMP by passing Int as an optional positional argument:

julia> @variable(model, x, Int)
x

Integer optimization variables can also be created by setting the integer keyword to true.

julia> @variable(model, x, integer=true)
x

We can check if an optimization variable is integer by calling is_integer on the JuMP variable, and integer constraints can be removed with unset_integer.

julia> is_integer(x)
true

julia> unset_integer(x)

julia> is_integer(x)
false

Semidefinite variables

JuMP also supports modeling with semidefinite variables. A square symmetric matrix $X$ is positive semidefinite if all eigenvalues are nonnegative. We can declare a matrix of JuMP variables to be positive semidefinite as follows:

julia> @variable(model, x[1:2, 1:2], PSD)
2×2 LinearAlgebra.Symmetric{VariableRef,Array{VariableRef,2}}:
 x[1,1]  x[1,2]
 x[1,2]  x[2,2]

Note that x must be a square 2-dimensional Array of JuMP variables; it cannot be a DenseAxisArray or a SparseAxisArray. (See Variable containers, above, for more on this.)

You can also impose a weaker constraint that the square matrix is only symmetric (instead of positive semidefinite) as follows:

julia> @variable(model, x[1:2, 1:2], Symmetric)
2×2 LinearAlgebra.Symmetric{VariableRef,Array{VariableRef,2}}:
 x[1,1]  x[1,2]
 x[1,2]  x[2,2]

Anonymous JuMP variables

In all of the above examples, we have created named JuMP variables. However, it is also possible to create so called anonymous JuMP variables. To create an anonymous JuMP variable, we drop the name of the variable from the macro call. This means dropping the second positional argument if the JuMP variable is a scalar, or dropping the name before the square bracket ([) if a container is being created. For example:

julia> x = @variable(model)
noname

This shows how (model, x) is really short for:

julia> x = model[:x] = @variable(model, base_name="x")
x

An Array of anonymous JuMP variables can be created as follows:

julia> y = @variable(model, [i=1:2])
2-element Array{VariableRef,1}:
 noname
 noname

If necessary, you can store x in model as follows:

julia> model[:x] = x

The <= and >= short-hand cannot be used to set bounds on anonymous JuMP variables. Instead, you should use the lower_bound and upper_bound keywords.

Passing the Bin and Int variable types are also invalid. Instead, you should use the binary and integer keywords.

Thus, the anonymous variant of @variable(model, x[i=1:2] >= i, Int) is:

julia> x = @variable(model, [i=1:2], base_name="x", lower_bound=i, integer=true)
2-element Array{VariableRef,1}:
 x[1]
 x[2]

User-defined containers

In the section Variable containers, we explained how JuMP supports the efficient creation of collections of JuMP variables in three types of containers. However, users are also free to create collections of JuMP variables in their own datastructures. For example, the following code creates a dictionary with symmetric matrices as the values:

julia> variables = Dict{Symbol, Array{VariableRef,2}}()
Dict{Symbol,Array{VariableRef,2}} with 0 entries

julia> for key in [:A, :B]
           global variables[key] = @variable(model, [1:2, 1:2])
       end

julia> variables
Dict{Symbol,Array{VariableRef,2}} with 2 entries:
  :A => VariableRef[noname noname; noname noname]
  :B => VariableRef[noname noname; noname noname]

Deleting variables

JuMP supports the deletion of optimization variables. To delete variables, we can use the delete method. We can also check whether x is a valid JuMP variable in model using the is_valid method:

julia> @variable(model, x)
x

julia> is_valid(model, x)
true

julia> delete(model, x)

julia> is_valid(model, x)
false

Listing all variables

Use JuMP.all_variables to obtain a list of all variables present in the model. This is useful for performing operations like:

• relaxing all integrality constraints in the model
• setting the starting values for variables to the result of the last solve

Reference

@variable(model, kw_args...)

@variable(model, expr, args..., kw_args...)

# Specify everything in `expr`
@variable(model, x >= 0)
# Specify the lower bound using a keyword argument
@variable(model, x, lower_bound=0)
# Specify everything in `kw_args`
x = @variable(model, base_name="x", lower_bound=0)

ub = Dict(:a => 2, :b => 3)
# Specify everything in `expr`
@variable(model, x[i=keys(ub)] <= ub[i])
# Specify the upper bound using a keyword argument
@variable(model, x[i=keys(ub)], upper_bound=ub[i])

info = VariableInfo(true, 0, false, NaN, false, NaN, false, NaN, false, false)
JuMP.add_variable(model, JuMP.build_variable(error, info), "x")

# Without the `@variable` macro
data = Vector{JuMP.variable_type(model)}(undef, length(keys(ub)))
x = JuMP.Containers.DenseAxisArray(data, keys(ub))
for i in keys(ub)
    info = VariableInfo(false, NaN, true, ub[i], false, NaN, false, NaN, false, false)
    x[i] = JuMP.add_variable(model, JuMP.build_variable(error, info), "x[$i]")
end

# Using the `@variable` macro
@variable(model, x[1:N,1:N], Symmetric, Poly(X))
# Without the `@variable` macro
x = Matrix{JuMP.variable_type(model, Poly(X))}(N, N)
info = VariableInfo(false, NaN, false, NaN, false, NaN, false, NaN, false, false)
for i in 1:N, j in i:N
    x[i,j] = x[j,i] = JuMP.add_variable(model, build_variable(error, info, Poly(X)), "x[$i,$j]")
end

owner_model(s::AbstractJuMPScalar)

VariableRef <: AbstractVariableRef

all_variables(model::Model)::Vector{VariableRef}

model = Model()
@variable(model, x)
@variable(model, y)
all_variables(model)

# output

2-element Array{VariableRef,1}:
 x
 y

has_lower_bound(v::VariableRef)

lower_bound(v::VariableRef)

set_lower_bound(v::VariableRef, lower::Number)

delete_lower_bound(v::VariableRef)

has_upper_bound(v::VariableRef)

upper_bound(v::VariableRef)

set_upper_bound(v::VariableRef,upper::Number)

delete_upper_bound(v::VariableRef)

is_fixed(v::VariableRef)

fix_value(v::VariableRef)

fix(v::VariableRef, value::Number; force::Bool = false)

unfix(v::VariableRef)

is_integer(v::VariableRef)

set_integer(variable_ref::VariableRef)

unset_integer(variable_ref::VariableRef)

IntegerRef(v::VariableRef)

is_binary(v::VariableRef)

set_binary(v::VariableRef)

unset_binary(variable_ref::VariableRef)

BinaryRef(v::VariableRef)

index(v::VariableRef)::MOI.VariableIndex

optimizer_index(v::VariableRef)::MOI.VariableIndex