CurrentModule = JuMP
DocTestSetup = quote
using JuMP
end
The term variable in mathematical optimization has many meanings. Here, we distinguish between the following three types of variables:
- optimization variables, which are the mathematical
xin the problem\max{f_0(x) | f_i(x) \in S_i}. - Julia variables, which are bindings between a name and a value, for example
x = 1. (See here for the Julia docs.) - JuMP variables, which are instances of the
JuMP.VariableRefstruct defined by JuMP that contains a reference to an optimization variable in a model. (Extra for experts: theVariableRefstruct is a thin wrapper around aMOI.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:
- it adds two optimization variables to
model - it creates two JuMP variables that act as references to those optimization variables
- 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.
!!! warn Creating two JuMP variables with the same name results in an error at runtime.
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:
- it adds one optimization variable to
model - it creates one JuMP variable that acts as a reference to that optimization variable
- it binds the JuMP variable to the Julia variable
y - it tells JuMP that the name attribute of this JuMP variable is "decision
variable". JuMP uses the value of
base_namewhen 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
# TODO(@odow): add a section on looking up by stringNow that we understand the difference between optimization, JuMP, and Julia
variables, we can introduce more of the functionality of the @variable macro.
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.
!!! note
When creating a variable with only a lower-bound or an upper-bound, and the
value of the bound is not a numeric literal, the name must appear on the
left-hand side. Putting the name on the right-hand side will result in an
error. For example:
julia @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 JuMP.has_lower_bound and JuMP.has_upper_bound functions. For example:
julia> JuMP.has_lower_bound(x_free)
false
julia> JuMP.has_upper_bound(x_upper)
true
If a variable has a lower or upper bound, we can query the value of it via the
JuMP.lower_bound and JuMP.upper_bound functions. For example:
julia> JuMP.lower_bound(x_interval)
2.0
julia> JuMP.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> JuMP.lower_bound(x)
1.0
Another option is to use the JuMP.set_lower_bound and JuMP.set_upper_bound
functions. These can also be used to modify an existing variable bound. For
example:
julia> @variable(model, x >= 1)
x
julia> JuMP.lower_bound(x)
1.0
julia> JuMP.set_lower_bound(x, 2)
julia> JuMP.lower_bound(x)
2.0
We can delete variable bounds using JuMP.delete_lower_bound and
JuMP.delete_upper_bound:
julia> @variable(model, 1 <= x <= 2)
x
julia> JuMP.lower_bound(x)
1.0
julia> JuMP.delete_lower_bound(x)
julia> JuMP.has_lower_bound(x)
false
julia> JuMP.upper_bound(x)
2.0
julia> JuMP.delete_upper_bound(x)
julia> JuMP.has_upper_bound(x)
false
In addition to upper and lower bounds, JuMP variables can also be fixed to a value.
julia> @variable(model, x == 1)
x
julia> JuMP.is_fixed(x)
true
julia> JuMP.fix_value(x)
1.0
julia> JuMP.unfix(x)
julia> JuMP.is_fixed(x)
false
Fixing a variable with existing bounds will throw an error. To delete the bounds
prior to fixing, use JuMP.fix(variable, value; force = true).
julia> @variable(model, x >= 1)
x
julia> JuMP.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> JuMP.fix(x, 2; force = true)
julia> JuMP.fix_value(x)
2.0
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(::JuMP.VariableRef)
set_name(::JuMP.VariableRef, ::String)
.
The variable can also be retrieved from its name using
JuMP.variable_by_name.
variable_by_name
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 datastructures.
JuMP provides a mechanism for creating three types of these datastructures,
which we refer to as containers. The three types are Arrays, DenseAxisArrays,
and SparseAxisArrays. We explain each of these in the following.
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]
We can also name each index, and 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> JuMP.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.
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]
DenseAxisArray's 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, the indices in a DenseAxisArray can be named,
and the bounds can depend upon these names. 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> JuMP.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
The third datatype that JuMP supports the efficient creation of are
SparseAxisArrays. These arrays are created when the indices do not form a
rectangular set. One example is 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]
x is a standard Julia dictionary. Therefore, slicing cannot be performed.
We can also conditionally create variables via a JuMP-specific syntax. This
sytax 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]
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 dictionary. 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 compile 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.
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 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 JuMP.is_binary on
the JuMP variable:
julia> JuMP.is_binary(x)
true
Binary optimization variables can also be created by setting the binary
keyword to true.
julia> @variable(model, x, binary=true)
x
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 JuMP.is_integer
on the JuMP variable:
julia> JuMP.is_integer(x)
true
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 JuMP array or a dictionary. (See Variable containers, above,
for more on this.)
You can also impose a slightly 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]
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]
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]
JuMP supports the deletion of optimization variables. To delete variables, we
can use the JuMP.delete method. We can also check whether x is a valid JuMP
variable in model using the JuMP.is_valid method:
julia> @variable(model, x)
x
julia> JuMP.is_valid(model, x)
true
julia> JuMP.delete(model, x)
julia> JuMP.is_valid(model, x)
false
@variable
owner_model