Getting started with Julia

Since JuMP is embedded in Julia, knowing some basic Julia is important for 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.

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

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:

@show 1 + 1
@show 1 - 2
@show 2 * 2
@show 4 / 5
@show 3^2

1 + 1 = 2
1 - 2 = -1
2 * 2 = 4
4 / 5 = 0.8
3 ^ 2 = 9

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:

@show typeof(1)
@show typeof(1.0)

typeof(1) = Int64
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.

We create complex numbers using im:

x = 2 + 1im
@show real(x)
@show imag(x)
@show typeof(x)
@show x * (1 - 2im)

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

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

π

π = 3.1415926535897...

typeof(π)

Irrational{:π}

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

typeof(2π / 3)

Float64

Floating point numbers

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:

0.1 * 3 == 0.3

false

sin(2π / 3) == √3 / 2

false

Let's see what the differences are:

0.1 * 3 - 0.3

5.551115123125783e-17

sin(2π / 3) - √3 / 2

1.1102230246251565e-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 ==:

0.1 * 3 ≈ 0.3

true

sin(2π / 3) ≈ √3 / 2

true

≈ is just a clever way of calling the isapprox function:

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

true

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

1 + 1e-16 == 1

true

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

(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:

b = [5, 6]

2-element Array{Int64,1}:
 5
 6

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

A = [1.0 2.0; 3.0 4.0]

2×2 Array{Float64,2}:
 1.0  2.0
 3.0  4.0

Note how this time the type is Array{Float64, 2}; the elements are Float64 and there are 2 dimensions.

We can do linear algebra:

x = A \ b

2-element Array{Float64,1}:
 -3.9999999999999987
  4.499999999999999 

A * x

2-element Array{Float64,1}:
 5.0
 6.0

A * x ≈ b

true

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

    b * b

MethodError: no method matching *(::Array{Int64,1}, ::Array{Int64,1})
Closest candidates are:
  *(::Any, ::Any, !Matched::Any, !Matched::Any...) at operators.jl:502
  *(!Matched::MutableArithmetics.Zero, ::Any) at /home/runner/.julia/packages/MutableArithmetics/8xkW3/src/rewrite.jl:49
  *(!Matched::ChainRulesCore.NotImplemented, ::Any) at /home/runner/.julia/packages/ChainRulesCore/Voykb/src/differential_arithmetic.jl:27
  ...

But multiplying transposes works:

b' * b

61

b * b'

2×2 Array{Int64,2}:
 25  30
 30  36

Other common types

Strings

Double quotes are used for strings:

typeof("This is Julia")

String

Unicode is fine in strings:

typeof("π is about 3.1415")

String

Use println to print a string:

println("Hello, World!")

Hello, World!

We can use $() to interpolate values into a string:

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

The value of x is: 123

Symbols

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

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

The value of x is: 123

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:

String(:abc)

"abc"

Symbol("abc")

:abc

Tuples

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

t = ("hello", 1.2, :foo)

("hello", 1.2, :foo)

typeof(t)

Tuple{String,Float64,Symbol}

Tuples can be accessed by index, similar to arrays:

t[2]

1.2

And they be "unpacked" like so:

a, b, c = t
b

1.2

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

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

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

Values can be accessed using dot syntax:

t.word

"hello"

Dictionaries

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:

d1 = Dict(1 => "A", 2 => "B", 4 => "D")

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

Looking up a values uses the bracket syntax:

d1[2]

"B"

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

Dict("A" => 1, "B" => 2.5, "D" => 2 - 3im)

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

Dict("A" => 1.0 + 0.0im, "B" => 2.5 + 0.0im, "D" => 2.0 - 3.0im)

Dict{String,Complex{Float64}} with 3 entries:
  "B" => 2.5+0.0im
  "A" => 1.0+0.0im
  "D" => 2.0-3.0im

Dictionaries can be nested:

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)

d2["B"]

2

d2["D"][:foo]

3

Loops

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

for i in 1:5
    println(i)
end

1
2
3
4
5

for i in [1.2, 2.3, 3.4, 4.5, 5.6]
    println(i)
end

1.2
2.3
3.4
4.5
5.6

This for-each loop also works with dictionaries:

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

B: 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:

for i in 0:3:15
    if i < 5
        println("$(i) is less than 5")
    elseif i < 10
        println("$(i) is less than 10")
    else
        if i == 10
            println("the value is 10")
        else
            println("$(i) is bigger than 10")
        end
    end
end

0 is less than 5
3 is less than 5
6 is less than 10
9 is less than 10
12 is bigger than 10
15 is bigger than 10

Comprehensions

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:

[i for i in 1:5]

5-element Array{Int64,1}:
 1
 2
 3
 4
 5

Matrices can be built by including multiple indices:

[i * j for i in 1:5, j in 5:10]

5×6 Array{Int64,2}:
  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:

[i for i in 1:10 if i % 2 == 1]

5-element Array{Int64,1}:
 1
 3
 5
 7
 9

A similar syntax can be used for building dictionaries:

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

Functions

A simple function is defined as follows:

function print_hello()
    return println("hello")
end
print_hello()

hello

Arguments can be added to a function:

function print_it(x)
    return println(x)
end
print_it("hello")
print_it(1.234)
print_it(:my_id)

hello
1.234
my_id

Optional keyword arguments are also possible:

function print_it(x; prefix = "value:")
    return println("$(prefix) $(x)")
end
print_it(1.234)
print_it(1.234, prefix = "val:")

value: 1.234
val: 1.234

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

function mult(x; y = 2.0)
    return x * y
end

mult(4.0)

8.0

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:

f = x -> x^2
f(2)

4

map(x -> x^2, 1:4)

4-element Array{Int64,1}:
  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:

function foo(x::Int)
    return x^2
end

function foo(x::Float64)
    return exp(x)
end

function foo(x::Number)
    return x + 1
end

@show foo(2)
@show foo(2.0)
@show foo(1 + 1im)

foo(2) = 4
foo(2.0) = 7.38905609893065
foo(1 + 1im) = 2 + 1im

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

    foo([1, 2, 3])

MethodError: no method matching foo(::Array{Int64,1})
Closest candidates are:
  foo(!Matched::Float64) at none:3
  foo(!Matched::Int64) at none:2
  foo(!Matched::Number) at none:3

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 Array{Int64, 1}, but it does know how to handle Float64, Int64, and Number.

Broadcasting

In the example above, we didn't define what to do if f was passed an Array. 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:

f.([1, 2, 3])

3-element Array{Int64,1}:
 1
 4
 9

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 can't modify the contents of a Float64.

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

function mutability_example(mutable_type::Vector{Int}, immutable_type::Int)
    mutable_type[1] += 1
    immutable_type += 1
    return
end

mutable_type = [1, 2, 3]
immutable_type = 1

mutability_example(mutable_type, immutable_type)

println("mutable_type: $(mutable_type)")
println("immutable_type: $(immutable_type)")

mutable_type: [2, 2, 3]
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:

isimmutable([1, 2, 3])

false

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.

using Random  # The equivalent of Python's `from Random import *`
import Random  # The equivalent of Python's `import Random`

Random.seed!(33)

[rand() for i in 1:10]

10-element Array{Float64,1}:
 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
Pkg.add("JuMP")

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
Pkg.add("https://github.com/user-name/MyPackage.jl.git")

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().

View this file on Github.

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