Containers

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JuMP provides specialized containers similar to AxisArrays that enable multi-dimensional arrays with non-integer indices.

These containers are created automatically by JuMP’s macros. Each macro has the same basic syntax:

@macroname(model, name[key1=index1, index2; optional_condition], other stuff)

The containers are generated by the name[key1=index1, index2; optional_condition] syntax. Everything else is specific to the particular macro.

Containers can be named, for example, name[key=index], or unnamed, for example, [key=index]. We call unnamed containers anonymous.

We call the bits inside the square brackets and before the ; the index sets. The index sets can be named, for example, [i = 1:4], or they can be unnamed, for example, [1:4].

We call the bit inside the square brackets and after the ; the condition. Conditions are optional.

In addition to the standard JuMP macros like @variable and @constraint, which construct containers of variables and constraints respectively, you can use Containers.@container to construct containers with arbitrary elements.

We will use this macro to explain the three types of containers that are natively supported by JuMP: Array, Containers.DenseAxisArray, and Containers.SparseAxisArray.

Array

An Array is created when the index sets are rectangular and the index sets are of the form 1:n.

julia> Containers.@container(x[i = 1:2, j = 1:3], (i, j))
2×3 Matrix{Tuple{Int64, Int64}}:
 (1, 1)  (1, 2)  (1, 3)
 (2, 1)  (2, 2)  (2, 3)

The result is a normal Julia Array, so you can do all the usual things.

Slicing

Arrays can be sliced

julia> x[:, 1]
2-element Vector{Tuple{Int64, Int64}}:
 (1, 1)
 (2, 1)

julia> x[2, :]
3-element Vector{Tuple{Int64, Int64}}:
 (2, 1)
 (2, 2)
 (2, 3)

Looping

Use eachindex to loop over the elements:

julia> for key in eachindex(x)
           println(x[key])
       end
(1, 1)
(2, 1)
(1, 2)
(2, 2)
(1, 3)
(2, 3)

Get the index sets

Use axes to obtain the index sets:

julia> axes(x)
(Base.OneTo(2), Base.OneTo(3))

Broadcasting

Broadcasting over an Array returns an Array

julia> swap(x::Tuple) = (last(x), first(x))
swap (generic function with 1 method)

julia> swap.(x)
2×3 Matrix{Tuple{Int64, Int64}}:
 (1, 1)  (2, 1)  (3, 1)
 (1, 2)  (2, 2)  (3, 2)

Tables

Use Containers.rowtable to convert the Array into a Tables.jl compatible Vector{<:NamedTuple}:

julia> table = Containers.rowtable(x; header = [:I, :J, :value])
6-element Vector{@NamedTuple{I::Int64, J::Int64, value::Tuple{Int64, Int64}}}:
 (I = 1, J = 1, value = (1, 1))
 (I = 2, J = 1, value = (2, 1))
 (I = 1, J = 2, value = (1, 2))
 (I = 2, J = 2, value = (2, 2))
 (I = 1, J = 3, value = (1, 3))
 (I = 2, J = 3, value = (2, 3))

Because it supports the Tables.jl interface, you can pass it to any function which accepts a table as input:

julia> import DataFrames;

julia> DataFrames.DataFrame(table)
6×3 DataFrame
 Row │ I      J      value
     │ Int64  Int64  Tuple…
─────┼──────────────────────
   1 │     1      1  (1, 1)
   2 │     2      1  (2, 1)
   3 │     1      2  (1, 2)
   4 │     2      2  (2, 2)
   5 │     1      3  (1, 3)
   6 │     2      3  (2, 3)

DenseAxisArray

A Containers.DenseAxisArray is created when the index sets are rectangular, but not of the form 1:n. The index sets can be of any type.

julia> x = Containers.@container([i = 1:2, j = [:A, :B]], (i, j))
2-dimensional DenseAxisArray{Tuple{Int64, Symbol},2,...} with index sets:
    Dimension 1, Base.OneTo(2)
    Dimension 2, [:A, :B]
And data, a 2×2 Matrix{Tuple{Int64, Symbol}}:
 (1, :A)  (1, :B)
 (2, :A)  (2, :B)

Slicing

DenseAxisArrays can be sliced

julia> x[:, :A]
1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:
    Dimension 1, Base.OneTo(2)
And data, a 2-element Vector{Tuple{Int64, Symbol}}:
 (1, :A)
 (2, :A)

julia> x[1, :]
1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:
    Dimension 1, [:A, :B]
And data, a 2-element Vector{Tuple{Int64, Symbol}}:
 (1, :A)
 (1, :B)

Looping

Use eachindex to loop over the elements:

julia> for key in eachindex(x)
           println(x[key])
       end
(1, :A)
(2, :A)
(1, :B)
(2, :B)

Get the index sets

Use axes to obtain the index sets:

julia> axes(x)
(Base.OneTo(2), [:A, :B])

Broadcasting

Broadcasting over a DenseAxisArray returns a DenseAxisArray

julia> swap(x::Tuple) = (last(x), first(x))
swap (generic function with 1 method)

julia> swap.(x)
2-dimensional DenseAxisArray{Tuple{Symbol, Int64},2,...} with index sets:
    Dimension 1, Base.OneTo(2)
    Dimension 2, [:A, :B]
And data, a 2×2 Matrix{Tuple{Symbol, Int64}}:
 (:A, 1)  (:B, 1)
 (:A, 2)  (:B, 2)

Access internal data

Use Array(x) to copy the internal data array into a new Array:

julia> Array(x)
2×2 Matrix{Tuple{Int64, Symbol}}:
 (1, :A)  (1, :B)
 (2, :A)  (2, :B)

To access the internal data without a copy, use x.data.

julia> x.data
2×2 Matrix{Tuple{Int64, Symbol}}:
 (1, :A)  (1, :B)
 (2, :A)  (2, :B)

Tables

Use Containers.rowtable to convert the DenseAxisArray into a Tables.jl compatible Vector{<:NamedTuple}:

julia> table = Containers.rowtable(x; header = [:I, :J, :value])
4-element Vector{@NamedTuple{I::Int64, J::Symbol, value::Tuple{Int64, Symbol}}}:
 (I = 1, J = :A, value = (1, :A))
 (I = 2, J = :A, value = (2, :A))
 (I = 1, J = :B, value = (1, :B))
 (I = 2, J = :B, value = (2, :B))

Because it supports the Tables.jl interface, you can pass it to any function which accepts a table as input:

julia> import DataFrames;

julia> DataFrames.DataFrame(table)
4×3 DataFrame
 Row │ I      J       value
     │ Int64  Symbol  Tuple…
─────┼────────────────────────
   1 │     1  A       (1, :A)
   2 │     2  A       (2, :A)
   3 │     1  B       (1, :B)
   4 │     2  B       (2, :B)

Keyword indexing

If all axes are named, you can use keyword indexing:

julia> x[i = 2, j = :A]
(2, :A)

julia> x[i = :, j = :B]
1-dimensional DenseAxisArray{Tuple{Int64, Symbol},1,...} with index sets:
    Dimension 1, Base.OneTo(2)
And data, a 2-element Vector{Tuple{Int64, Symbol}}:
 (1, :B)
 (2, :B)

SparseAxisArray

A Containers.SparseAxisArray is created when the index sets are non-rectangular. This occurs in two circumstances:

An index depends on a prior index:

julia> Containers.@container([i = 1:2, j = i:2], (i, j))
JuMP.Containers.SparseAxisArray{Tuple{Int64, Int64}, 2, Tuple{Int64, Int64}} with 3 entries:
  [1, 1]  =  (1, 1)
  [1, 2]  =  (1, 2)
  [2, 2]  =  (2, 2)

The [indices; condition] syntax is used:

julia> x = Containers.@container([i = 1:3, j = [:A, :B]; i > 1], (i, j))
JuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 2, Tuple{Int64, Symbol}} with 4 entries:
  [2, A]  =  (2, :A)
  [2, B]  =  (2, :B)
  [3, A]  =  (3, :A)
  [3, B]  =  (3, :B)

Here we have the index sets i = 1:3, j = [:A, :B], followed by ;, and then a condition, which evaluates to true or false: i > 1.

Slicing

Slicing is supported:

julia> y = x[:, :B]
JuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 1, Tuple{Int64}} with 2 entries:
  [2]  =  (2, :B)
  [3]  =  (3, :B)

Looping

Use eachindex to loop over the elements:

julia> for key in eachindex(x)
           println(x[key])
       end
(2, :A)
(2, :B)
(3, :A)
(3, :B)

julia> for key in eachindex(y)
           println(y[key])
       end
(2, :B)
(3, :B)

If you use a macro to construct a SparseAxisArray, then the iteration order is row-major, that is, indices are varied from right to left. As an example, when iterating over x above, the j index is iterated, keeping i constant. This order is in contrast to Base.Arrays, which iterate in column-major order, that is, by varying indices from left to right.

Broadcasting

Broadcasting over a SparseAxisArray returns a SparseAxisArray

julia> swap(x::Tuple) = (last(x), first(x))
swap (generic function with 1 method)

julia> swap.(y)
JuMP.Containers.SparseAxisArray{Tuple{Symbol, Int64}, 1, Tuple{Int64}} with 2 entries:
  [2]  =  (:B, 2)
  [3]  =  (:B, 3)

Tables

Use Containers.rowtable to convert the SparseAxisArray into a Tables.jl compatible Vector{<:NamedTuple}:

julia> table = Containers.rowtable(x; header = [:I, :J, :value])
4-element Vector{@NamedTuple{I::Int64, J::Symbol, value::Tuple{Int64, Symbol}}}:
 (I = 2, J = :A, value = (2, :A))
 (I = 2, J = :B, value = (2, :B))
 (I = 3, J = :A, value = (3, :A))
 (I = 3, J = :B, value = (3, :B))

Because it supports the Tables.jl interface, you can pass it to any function which accepts a table as input:

julia> import DataFrames;

julia> DataFrames.DataFrame(table)
4×3 DataFrame
 Row │ I      J       value
     │ Int64  Symbol  Tuple…
─────┼────────────────────────
   1 │     2  A       (2, :A)
   2 │     2  B       (2, :B)
   3 │     3  A       (3, :A)
   4 │     3  B       (3, :B)

Keyword indexing

If all axes are named, you can use keyword indexing:

julia> x[i = 2, j = :A]
(2, :A)

julia> x[i = :, j = :B]
JuMP.Containers.SparseAxisArray{Tuple{Int64, Symbol}, 1, Tuple{Int64}} with 2 entries:
  [2]  =  (2, :B)
  [3]  =  (3, :B)

Forcing the container type

Pass container = T to use T as the container. For example:

julia> Containers.@container([i = 1:2, j = 1:2], i + j, container = Array)
2×2 Matrix{Int64}:
 2  3
 3  4

julia> Containers.@container([i = 1:2, j = 1:2], i + j, container = Dict)
Dict{Tuple{Int64, Int64}, Int64} with 4 entries:
  (1, 2) => 3
  (1, 1) => 2
  (2, 2) => 4
  (2, 1) => 3

You can also pass DenseAxisArray or SparseAxisArray.

How different container types are chosen

If the compiler can prove at compile time that the index sets are rectangular, and indexed by a compact set of integers that start at 1, Containers.@container will return an array. This is the case if your index sets are visible to the macro as 1:n:

julia> Containers.@container([i=1:3, j=1:5], i + j)
3×5 Matrix{Int64}:
 2  3  4  5  6
 3  4  5  6  7
 4  5  6  7  8

or an instance of Base.OneTo:

julia> set = Base.OneTo(3)
Base.OneTo(3)

julia> Containers.@container([i=set, j=1:5], i + j)
3×5 Matrix{Int64}:
 2  3  4  5  6
 3  4  5  6  7
 4  5  6  7  8

If the compiler can prove that the index set is rectangular, but not necessarily of the form 1:n at compile time, then a Containers.DenseAxisArray will be constructed instead:

julia> set = 1:3
1:3

julia> Containers.@container([i=set, j=1:5], i + j)
2-dimensional DenseAxisArray{Int64,2,...} with index sets:
    Dimension 1, 1:3
    Dimension 2, Base.OneTo(5)
And data, a 3×5 Matrix{Int64}:
 2  3  4  5  6
 3  4  5  6  7
 4  5  6  7  8

What happened here? Although we know that set contains 1:3, at compile time the typeof(set) is a UnitRange{Int}. Therefore, Julia can’t prove that the range starts at 1 (it only finds this out at runtime), and it defaults to a DenseAxisArray. The case where we explicitly wrote i = 1:3 worked because the macro can "see" the 1 at compile time.

However, if you know that the indices do form an Array, you can force the container type with container = Array:

julia> set = 1:3
1:3

julia> Containers.@container([i=set, j=1:5], i + j, container = Array)
3×5 Matrix{Int64}:
 2  3  4  5  6
 3  4  5  6  7
 4  5  6  7  8

Here’s another example with something similar:

julia> a = 1
1

julia> Containers.@container([i=a:3, j=1:5], i + j)
2-dimensional DenseAxisArray{Int64,2,...} with index sets:
    Dimension 1, 1:3
    Dimension 2, Base.OneTo(5)
And data, a 3×5 Matrix{Int64}:
 2  3  4  5  6
 3  4  5  6  7
 4  5  6  7  8

julia> Containers.@container([i=1:a, j=1:5], i + j)
1×5 Matrix{Int64}:
 2  3  4  5  6

Finally, if the compiler cannot prove that the index set is rectangular, a Containers.SparseAxisArray will be created.

This occurs when some indices depend on a previous one:

julia> Containers.@container([i=1:3, j=1:i], i + j)
JuMP.Containers.SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 6 entries:
  [1, 1]  =  2
  [2, 1]  =  3
  [2, 2]  =  4
  [3, 1]  =  4
  [3, 2]  =  5
  [3, 3]  =  6

or if there is a condition on the index sets:

julia> Containers.@container([i = 1:5; isodd(i)], i^2)
JuMP.Containers.SparseAxisArray{Int64, 1, Tuple{Int64}} with 3 entries:
  [1]  =  1
  [3]  =  9
  [5]  =  25

The condition can depend on multiple indices, the only requirement is that it is an expression that returns true or false:

julia> condition(i, j) = isodd(i) && iseven(j)
condition (generic function with 1 method)

julia> Containers.@container([i = 1:2, j = 1:4; condition(i, j)], i + j)
JuMP.Containers.SparseAxisArray{Int64, 2, Tuple{Int64, Int64}} with 2 entries:
  [1, 2]  =  3
  [1, 4]  =  5