# 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:

*optimization*variables, which are the mathematical $x$ in 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`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:

- 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.

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_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.

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 of the variable must appear on the left-hand side. Putting the name on the right-hand side will result in an error. For example:

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

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)`

.

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

.

## 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 `Array`

s, `DenseAxisArray`

s, and `SparseAxisArray`

s. 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, Base.OneTo(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{Int64,Int64}} 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{Int64}} with 2 entries:
[4] = x[4]
[2] = x[2]
```

Note that with many index dimensions and a large amount of sparsity, variable construction may be unnecessarily slow if the semi-colon syntax is naively applied. When using the semi-colon as a filter, JuMP iterates over *all* indices and evaluates the conditional for each combination. When this is undesired, the recommended work-around is to work directly with a list of tuples or create a dictionary. Consider the following examples:

```
N = 10
S = [(1, 1, 1),(N, N, N)]
# Slow. It evaluates conditional N^3 times.
@variable(model, x1[i=1:N, j=1:N, k=1:N; (i, j, k) in S])
# Fast.
@variable(model, x2[S])
# Fast. Manually constructs a dictionary and fills it.
x3 = Dict()
for (i, j, k) in S
x3[i, j, k] = @variable(model)
# Optional, if you care about pretty printing:
set_name(x3[i, j, k], "x[$i,$j,$k]")
end
```

### 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 utilities

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
```

### Relaxing integrality

The `relax_integrality`

function relaxes all integrality constraints in the model, returning a function that can be called to undo the operation later on.

## 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]
```

or using the syntax for Variables constrained on creation:

```
julia> @variable(model, x[1:2, 1:2] in PSDCone())
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]
```

You can impose a constraint that the square matrix is skew symmetric with `SkewSymmetricMatrixSpace`

:

```
julia> @variable(model, x[1:2, 1:2] in SkewSymmetricMatrixSpace())
2×2 Array{GenericAffExpr{Float64,VariableRef},2}:
0 x[1,2]
-x[1,2] 0
```

## 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 `@variable(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]
```

Creating two named JuMP variables with the same name results in an error at runtime. Use anonymous variables as an alternative.

## Variables constrained on creation

When using JuMP in Direct mode, it may be required to constrain variables on creation instead of constraining free variables as the solver may only support variables constrained on creation. In Automatic and Manual modes, both ways of adding constraints on variables are equivalent. Indeed, during the copy of the cache to the optimizer, the choice of the constraints on variables that are copied as variables constrained on creation does not depend on how it was added to the cache.

All uses of the `@variable`

macro documented so far translate to a separate call for variable creation and adding of constraints.

For example, `@variable(model, x >= 0, Int)`

, is equivalent to:

```
@variable(model, x)
set_lower_bound(x, 0.0)
@constraint(model, x in MOI.Integer())
```

Importantly, the bound and integrality constraints are added *after* the variable has been created.

However, some solvers require a constraining set *at creation time*. We say that these variables are *constrained on creation*.

Use `in`

within `@variable`

to access the special syntax for constraining variables on creation. For example, the following creates a vector of variables constrained on creation to belong to the `SecondOrderCone`

:

```
julia> @variable(model, y[1:3] in SecondOrderCone())
3-element Array{VariableRef,1}:
y[1]
y[2]
y[3]
```

For contrast, the more standard approach is as follows:

```
julia> @variable(model, x[1:3])
3-element Array{VariableRef,1}:
x[1]
x[2]
x[3]
julia> @constraint(model, x in SecondOrderCone())
[x[1], x[2], x[3]] ∈ MathOptInterface.SecondOrderCone(3)
```

The technical difference between the former and the latter is that the former calls `MOI.add_constrained_variables`

while the latter calls `MOI.add_variables`

and then `MOI.add_constraint`

. This distinction is important only in Direct mode, depending on the solver being used. It's often not possible to delete the `SecondOrderCone`

constraint if it was specified at variable creation time.

### The `set`

keyword

An alternate syntax to `x in Set`

is to use the `set`

keyword of `@variable`

. This is most useful when creating anonymous variables:

`x = @variable(model, [1:2, 1:2], set = PSDCone())`

## 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

## Start values

There are two ways to provide a primal starting solution (also called MIP-start or a warmstart) for each variable:

- using the
`start`

keyword in the`@variable`

macro - using
`set_start_value`

The starting value of a variable can be queried using `start_value`

. If no start value has been set, `start_value`

will return `nothing`

.

```
julia> @variable(model, x)
x
julia> start_value(x)
julia> @variable(model, y, start = 1)
y
julia> start_value(y)
1.0
julia> set_start_value(y, 2)
julia> start_value(y)
2.0
```

Prior to JuMP 0.19, the previous solution to a solve was automatically set as the new starting value. JuMP 0.19 no longer does this automatically. To reproduce the functionality, use:

`set_start_value.(all_variables(model), value.(all_variables(model)))`

## The `@variables`

macro

If you have many `@variable`

calls, JuMP provides the macro `@variables`

that can improve readability:

```
julia> @variables(model, begin
x
y[i=1:2] >= i, (start = i, base_name = "Y_$i")
z, Bin
end)
julia> print(model)
Feasibility
Subject to
Y_1[1] ≥ 1.0
Y_2[2] ≥ 2.0
z binary
```

Keyword arguments must be contained within parentheses. (See the example above.)