Getting started with Julia

Because JuMP is embedded in Julia, knowing some basic Julia is important before you start learning JuMP.


This tutorial is designed to provide a minimalist crash course in the basics of Julia. You can find resources that provide a more comprehensive introduction to Julia here.

Installing Julia

To install Julia, download the latest stable release, then follow the platform specific install instructions.


Unless you know otherwise, you probably want the 64-bit version.

Next, you need an IDE to develop in. VS Code is a popular choice, so follow these install instructions.

Julia can also be used with Jupyter notebooks or the reactive notebooks of Pluto.jl.

The Julia REPL

The main way of interacting with Julia is via its REPL (Read Evaluate Print Loop). To access the REPL, start the Julia executable to arrive at the julia> prompt, and then start coding:

julia> 1 + 12

As your programs become larger, write a script as a text file, and then run that file using:

julia> include("path/to/file.jl")

Because of Julia's startup latency, running scripts from the command line like the following is slow:

$ julia path/to/file.jl

Use the REPL or a notebook instead, and read The "time-to-first-solve" issue for more information.

Code blocks in this documentation

In this documentation you'll see a mix of code examples with and without the julia>.

The Julia prompt is mostly used to demonstrate short code snippets, and the output is exactly what you will see if run from the REPL.

Blocks without the julia> can be copy-pasted into the REPL, but they are used because they enable richer output like plots or LaTeX to be displayed in the online and PDF versions of the documentation. If you run them from the REPL you may see different output.

Where to get help

To access the built-in help at the REPL, type ? to enter help-mode, followed by the name of the function to lookup:

help?> print
search: print println printstyled sprint isprint prevind parentindices precision escape_string

  print([io::IO], xs...)

  Write to io (or to the default output stream stdout if io is not given) a canonical
  (un-decorated) text representation. The representation used by print includes minimal formatting
  and tries to avoid Julia-specific details.

  print falls back to calling show, so most types should just define show. Define print if your
  type has a separate "plain" representation. For example, show displays strings with quotes, and
  print displays strings without quotes.

  string returns the output of print as a string.


  julia> print("Hello World!")
  Hello World!
  julia> io = IOBuffer();

  julia> print(io, "Hello", ' ', :World!)

  julia> String(take!(io))
  "Hello World!"

Numbers and arithmetic

Since we want to solve optimization problems, we're going to be using a lot of math. Luckily, Julia is great for math, with all the usual operators:

julia> 1 + 12
julia> 1 - 2-1
julia> 2 * 24
julia> 4 / 50.8
julia> 3^29

Did you notice how Julia didn't print .0 after some of the numbers? Julia is a dynamic language, which means you never have to explicitly declare the type of a variable. However, in the background, Julia is giving each variable a type. Check the type of something using the typeof function:

julia> typeof(1)Int64
julia> typeof(1.0)Float64

Here 1 is an Int64, which is an integer with 64 bits of precision, and 1.0 is a Float64, which is a floating point number with 64-bits of precision.


If you aren't familiar with floating point numbers, make sure to read the Floating point numbers section.

We create complex numbers using im:

julia> x = 2 + 1im2 + 1im
julia> real(x)2
julia> imag(x)1
julia> typeof(x)Complex{Int64}
julia> x * (1 - 2im)4 - 3im

The curly brackets surround what we call the parameters of a type. You can read Complex{Int64} as "a complex number, where the real and imaginary parts are represented by Int64." If we call typeof(1.0 + 2.0im) it will be Complex{Float64}, which a complex number with the parts represented by Float64.

There are also some cool things like an irrational representation of π.

julia> ππ = 3.1415926535897...

To make π (and most other Greek letters), type \pi and then press [TAB].

julia> typeof(π)Irrational{:π}

However, if we do math with irrational numbers, they get converted to Float64:

julia> typeof(2π / 3)Float64

Floating point numbers


If you aren't familiar with floating point numbers, make sure to read this section carefully.

A Float64 is a floating point approximation of a real number using 64-bits of information.

Because it is an approximation, things we know hold true in mathematics don't hold true in a computer! For example:

julia> 0.1 * 3 == 0.3false

A more complicated example is:

julia> sin(2π / 3) == √3 / 2false

Get by typing \sqrt then press [TAB].

Let's see what the differences are:

julia> 0.1 * 3 - 0.35.551115123125783e-17
julia> sin(2π / 3) - √3 / 21.1102230246251565e-16

They are small, but not zero!

One way of explaining this difference is to consider how we would write 1 / 3 and 2 / 3 using only four digits after the decimal point. We would write 1 / 3 as 0.3333, and 2 / 3 as 0.6667. So, despite the fact that 2 * (1 / 3) == 2 / 3, 2 * 0.3333 == 0.6666 != 0.6667.

Let's try that again using ≈ (\approx + [TAB]) instead of ==:

julia> 0.1 * 3 ≈ 0.3true
julia> sin(2π / 3) ≈ √3 / 2true

is a clever way of calling the isapprox function:

julia> isapprox(sin(2π / 3), √3 / 2; atol = 1e-8)true

Floating point is the reason solvers use tolerances when they solve optimization models. A common mistake you're likely to make is checking whether a binary variable is 0 using value(z) == 0. Always remember to use something like isapprox when comparing floating point numbers.

Note that isapprox will always return false if one of the number being compared is 0 and atol is zero (its default value).

julia> 1e-300 ≈ 0.0false

so always set a nonzero value of atol if one of the arguments can be zero.

julia> isapprox(1e-9, 0.0; atol = 1e-8)true

Gurobi has a good series of articles on the implications of floating point in optimization if you want to read more.

If you aren't careful, floating point arithmetic can throw up all manner of issues. For example:

julia> 1 + 1e-16 == 1true

It even turns out that floating point numbers aren't associative!

julia> (1 + 1e-16) - 1e-16 == 1 + (1e-16 - 1e-16)false

It's important to note that this issue isn't Julia-specific. It happens in every programming language (try it out in Python).

Vectors, matrices and arrays

Similar to Matlab, Julia has native support for vectors, matrices and tensors; all of which are represented by arrays of different dimensions. Vectors are constructed by comma-separated elements surrounded by square brackets:

julia> b = [5, 6]2-element Vector{Int64}:

Matrices can be constructed with spaces separating the columns, and semicolons separating the rows:

julia> A = [1.0 2.0; 3.0 4.0]2×2 Matrix{Float64}:
 1.0  2.0
 3.0  4.0

We can do linear algebra:

julia> x = A \ b2-element Vector{Float64}:

Here is floating point at work again! x is approximately [-4, 4.5].

julia> A * x2-element Vector{Float64}:
julia> A * x ≈ btrue

Note that when multiplying vectors and matrices, dimensions matter. For example, you can't multiply a vector by a vector:

julia>     b * bMethodError: no method matching *(::Vector{Int64}, ::Vector{Int64})
Closest candidates are:
  *(::Any, ::Any, !Matched::Any, !Matched::Any...) at operators.jl:560
  *(!Matched::StridedMatrix{T}, ::StridedVector{S}) where {T<:Union{Float32, Float64, ComplexF32, ComplexF64}, S<:Real} at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/matmul.jl:44
  *(::StridedVecOrMat{T} where T, !Matched::LinearAlgebra.Adjoint{var"#s814", var"#s813"} where {var"#s814", var"#s813"<:LinearAlgebra.LQPackedQ}) at /buildworker/worker/package_linux64/build/usr/share/julia/stdlib/v1.6/LinearAlgebra/src/lq.jl:254

But multiplying transposes works:

julia> b' * b61
julia> b * b'2×2 Matrix{Int64}: 25 30 30 36

Other common types


Although not technically a type, code comments begin with the # character:

julia> 1 + 1  # This is a comment2

Multiline comments begin with #= and end with =#:

Here is a
multiline comment

Comments can even be nested inside expressions. This is sometimes helpful when documenting inputs to functions:

julia> isapprox(
           #= We need an explicit atol here because we are comparing with 0 =#
           atol = 0.001,


Double quotes are used for strings:

julia> typeof("This is Julia")String

Unicode is fine in strings:

julia> typeof("π is about 3.1415")String

Use println to print a string:

julia> println("Hello, World!")Hello, World!

Use $() to interpolate values into a string:

julia> x = 123123
julia> println("The value of x is: $(x)")The value of x is: 123

Use triple-quotes for multiline strings:

julia> s = """
       Here is
       multiline string
       """"Here is\na\nmultiline string\n"
julia> println(s)Here is a multiline string


Julia Symbols are a data structure from the compiler that represent Julia identifiers (i.e., variable names).

julia> println("The value of x is: $(eval(:x))")The value of x is: 123

We used eval here to demonstrate how Julia links Symbols to variables. However, avoid calling eval in your code. It is usually a sign that your code is doing something that could be more easily achieved a different way. The Community Forum is a good place to ask for advice on alternative approaches.

julia> typeof(:x)Symbol

You can think of a Symbol as a String that takes up less memory, and that can't be modified.

Convert between String and Symbol using their constructors:

julia> String(:abc)"abc"
julia> Symbol("abc"):abc

Symbols are often (ab)used to stand in for a String or an Enum, when one of the latter is likely a better choice. The JuMP Style guide recommends reserving Symbols for identifiers. See @enum vs. Symbol for more.


Julia makes extensive use of a simple data structure called Tuples. Tuples are immutable collections of values. For example:

julia> t = ("hello", 1.2, :foo)("hello", 1.2, :foo)
julia> typeof(t)Tuple{String, Float64, Symbol}

Tuples can be accessed by index, similar to arrays:

julia> t[2]1.2

And they can be "unpacked" like so:

julia> a, b, c = t("hello", 1.2, :foo)
julia> b1.2

The values can also be given names, which is a convenient way of making light-weight data structures.

julia> t = (word = "hello", num = 1.2, sym = :foo)(word = "hello", num = 1.2, sym = :foo)

Values can be accessed using dot syntax:

julia> t.word"hello"


Similar to Python, Julia has native support for dictionaries. Dictionaries provide a very generic way of mapping keys to values. For example, a map of integers to strings:

julia> d1 = Dict(1 => "A", 2 => "B", 4 => "D")Dict{Int64, String} with 3 entries:
  4 => "D"
  2 => "B"
  1 => "A"

Type-stuff again: Dict{Int64,String} is a dictionary with Int64 keys and String values.

Looking up a value uses the bracket syntax:

julia> d1[2]"B"

Dictionaries support non-integer keys and can mix data types:

julia> Dict("A" => 1, "B" => 2.5, "D" => 2 - 3im)Dict{String, Number} with 3 entries:
  "B" => 2.5
  "A" => 1
  "D" => 2-3im

Julia types form a hierarchy. Here the value type of the dictionary is Number, which is a generalization of Int64, Float64, and Complex{Int}. Leaf nodes in this hierarchy are called "concrete" types, and all others are called "Abstract." In general, having variables with abstract types like Number can lead to slower code, so you should try to make sure every element in a dictionary or vector is the same type. For example, in this case we could represent every element as a Complex{Float64}:

julia> Dict("A" => 1.0 + 0.0im, "B" => 2.5 + 0.0im, "D" => 2.0 - 3.0im)Dict{String, ComplexF64} with 3 entries:
  "B" => 2.5+0.0im
  "A" => 1.0+0.0im
  "D" => 2.0-3.0im

Dictionaries can be nested:

julia> d2 = Dict("A" => 1, "B" => 2, "D" => Dict(:foo => 3, :bar => 4))Dict{String, Any} with 3 entries:
  "B" => 2
  "A" => 1
  "D" => Dict(:bar=>4, :foo=>3)
julia> d2["B"]2
julia> d2["D"][:foo]3


You can define custom datastructures with struct:

julia> struct MyStruct
julia> a = MyStruct(1, "a", Dict(2 => 3))Main.MyStruct(1, "a", Dict(2 => 3))
julia> a.x1

By default, these are not mutable

julia>     a.x = 2setfield! immutable struct of type MyStruct cannot be changed

However, you can declare a mutable struct which is mutable:

julia> mutable struct MyStructMutable
julia> a = MyStructMutable(1, "a", Dict(2 => 3))Main.MyStructMutable(1, "a", Dict(2 => 3))
julia> a.x1
julia> a.x = 22
julia> aMain.MyStructMutable(2, "a", Dict(2 => 3))


Julia has native support for for-each style loops with the syntax for <value> in <collection> end:

julia> for i in 1:5

Ranges are constructed as start:stop, or start:step:stop.

julia> for i in 1.2:1.1:5.6

This for-each loop also works with dictionaries:

julia> for (key, value) in Dict("A" => 1, "B" => 2.5, "D" => 2 - 3im)
           println("$(key): $(value)")
       endB: 2.5
A: 1
D: 2 - 3im

Note that in contrast to vector languages like Matlab and R, loops do not result in a significant performance degradation in Julia.

Control flow

Julia control flow is similar to Matlab, using the keywords if-elseif-else-end, and the logical operators || and && for or and and respectively:

julia> for i in 0:5:15
           if i < 5
               println("$(i) is less than 5")
           elseif i < 10
               println("$(i) is less than 10")
               if i == 10
                   println("the value is 10")
                   println("$(i) is bigger than 10")
       end0 is less than 5
5 is less than 10
the value is 10
15 is bigger than 10


Similar to languages like Haskell and Python, Julia supports the use of simple loops in the construction of arrays and dictionaries, called comprehensions.

A list of increasing integers:

julia> [i for i in 1:5]5-element Vector{Int64}:

Matrices can be built by including multiple indices:

julia> [i * j for i in 1:5, j in 5:10]5×6 Matrix{Int64}:
  5   6   7   8   9  10
 10  12  14  16  18  20
 15  18  21  24  27  30
 20  24  28  32  36  40
 25  30  35  40  45  50

Conditional statements can be used to filter out some values:

julia> [i for i in 1:10 if i % 2 == 1]5-element Vector{Int64}:

A similar syntax can be used for building dictionaries:

julia> Dict("$(i)" => i for i in 1:10 if i % 2 == 1)Dict{String, Int64} with 5 entries:
  "1" => 1
  "5" => 5
  "7" => 7
  "9" => 9
  "3" => 3


A simple function is defined as follows:

julia> function print_hello()
           return println("hello")
       endprint_hello (generic function with 1 method)
julia> print_hello()hello

Arguments can be added to a function:

julia> function print_it(x)
           return println(x)
       endprint_it (generic function with 1 method)
julia> print_it("hello")hello
julia> print_it(1.234)1.234
julia> print_it(:my_id)my_id

Optional keyword arguments are also possible:

julia> function print_it(x; prefix = "value:")
           return println("$(prefix) $(x)")
       endprint_it (generic function with 1 method)
julia> print_it(1.234)value: 1.234
julia> print_it(1.234; prefix = "val:")val: 1.234

The keyword return is used to specify the return values of a function:

julia> function mult(x; y = 2.0)
           return x * y
       endmult (generic function with 1 method)
julia> mult(4.0)8.0
julia> mult(4.0; y = 5.0)20.0

Anonymous functions

The syntax input -> output creates an anonymous function. These are most useful when passed to other functions. For example:

julia> f = x -> x^2#11 (generic function with 1 method)
julia> f(2)4
julia> map(x -> x^2, 1:4)4-element Vector{Int64}: 1 4 9 16

Type parameters

We can constrain the inputs to a function using type parameters, which are :: followed by the type of the input we want. For example:

julia> function foo(x::Int)
           return x^2
       endfoo (generic function with 1 method)
julia> function foo(x::Float64) return exp(x) endfoo (generic function with 2 methods)
julia> function foo(x::Number) return x + 1 endfoo (generic function with 3 methods)
julia> foo(2)4
julia> foo(2.0)7.38905609893065
julia> foo(1 + 1im)2 + 1im

But what happens if we call foo with something we haven't defined it for?

julia>     foo([1, 2, 3])MethodError: no method matching foo(::Vector{Int64})
Closest candidates are:
  foo(!Matched::Int64) at REPL[1]:1
  foo(!Matched::Float64) at REPL[2]:1
  foo(!Matched::Number) at REPL[3]:1

We get a dreaded MethodError! A MethodError means that you passed a function something that didn't match the type that it was expecting. In this case, the error message says that it doesn't know how to handle an Vector{Int64}, but it does know how to handle Float64, Int64, and Number.


Read the "Closest candidates" part of the error message carefully to get a hint as to what was expected.


In the example above, we didn't define what to do if f was passed a Vector. Luckily, Julia provides a convenient syntax for mapping f element-wise over arrays! Just add a . between the name of the function and the opening (. This works for any function, including functions with multiple arguments. For example:

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

Get a MethodError when calling a function that takes a Vector, Matrix, or Array? Try broadcasting it!

Mutable vs immutable objects

Some types in Julia are mutable, which means you can change the values inside them. A good example is an array. You can modify the contents of an array without having to make a new array.

In contrast, types like Float64 are immutable. You cannot modify the contents of a Float64.

This is something to be aware of when passing types into functions. For example:

julia> function mutability_example(mutable_type::Vector{Int}, immutable_type::Int)
           mutable_type[1] += 1
           immutable_type += 1
       endmutability_example (generic function with 1 method)
julia> mutable_type = [1, 2, 3]3-element Vector{Int64}: 1 2 3
julia> immutable_type = 11
julia> mutability_example(mutable_type, immutable_type)
julia> println("mutable_type: $(mutable_type)")mutable_type: [2, 2, 3]
julia> println("immutable_type: $(immutable_type)")immutable_type: 1

Because Vector{Int} is a mutable type, modifying the variable inside the function changed the value outside of the function. In contrast, the change to immutable_type didn't modify the value outside the function.

You can check mutability with the isimmutable function:

julia> isimmutable([1, 2, 3])false
julia> isimmutable(1)true

The package manager

Installing packages

No matter how wonderful Julia's base language is, at some point you will want to use an extension package. Some of these are built-in, for example random number generation is available in the Random package in the standard library. These packages are loaded with the commands using and import.

julia> using Random  # The equivalent of Python's `from Random import *`
julia> import Random # The equivalent of Python's `import Random`
julia> Random.seed!(33)MersenneTwister(33)
julia> [rand() for i in 1:10]10-element Vector{Float64}: 0.8245577112736127 0.2928364052074266 0.8765793121770682 0.41615145984974955 0.7113242552761618 0.7762718106176869 0.407423649552187 0.15761624576044575 0.8889767003637221 0.017829104289712516

The Package Manager is used to install packages that are not part of Julia's standard library.

For example the following can be used to install JuMP,

using Pkg

For a complete list of registered Julia packages see the package listing at JuliaHub.

From time to you may wish to use a Julia package that is not registered. In this case a git repository URL can be used to install the package.

using Pkg

Package environments

By default, Pkg.add will add packages to Julia's global environment. However, Julia also has built-in support for virtual environments.

Activate a virtual environment with:

import Pkg; Pkg.activate("/path/to/environment")

You can see what packages are installed in the current environment with Pkg.status().


We strongly recommend you create a Pkg environment for each project that you create in Julia, and add only the packages that you need, instead of adding lots of packages to the global environment. The Pkg manager documentation has more information on this topic.


This tutorial was generated using Literate.jl. View the source .jl file on GitHub.