Getting started with JuMP

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This tutorial is aimed at providing a quick introduction to writing and solving optimization models with JuMP.

If you’re new to Julia, start by reading Getting started with Julia.

What is JuMP?

JuMP ("Julia for Mathematical Programming") is an open-source modeling language that is embedded in Julia. It allows users to formulate various classes of optimization problems (linear, mixed-integer, quadratic, conic quadratic, semidefinite, and nonlinear) with easy-to-read code. These problems can then be solved using state-of-the-art open-source and commercial solvers.

JuMP also makes advanced optimization techniques easily accessible from a high-level language.

What is a solver?

A solver is a software package that incorporates algorithms for finding solutions to one or more classes of problem.

For example, HiGHS is a solver for linear programming (LP) and mixed integer programming (MIP) problems. It incorporates algorithms such as the simplex method and the interior-point method.

The Supported-solvers table lists the open-source and commercial solvers that JuMP currently supports.

What is MathOptInterface?

Each solver has its own concepts and data structures for representing optimization models and obtaining results.

MathOptInterface (MOI) is an abstraction layer that JuMP uses to convert from the problem written in JuMP to the solver-specific data structures for each solver.

MOI can be used directly, or through a higher-level modeling interface like JuMP.

Because JuMP is built on top of MOI, you’ll often see the MathOptInterface. prefix displayed when JuMP types are printed. However, you’ll only need to understand and interact with MOI to accomplish advanced tasks such as creating solver-independent callbacks.

Installation

JuMP is a package for Julia. From Julia, JuMP is installed by using the built-in package manager.

import Pkg
Pkg.add("JuMP")

You also need to include a Julia package which provides an appropriate solver. One such solver is HiGHS.Optimizer, which is provided by the HiGHS.jl package.

import Pkg
Pkg.add("HiGHS")

See Installation Guide for a list of other solvers you can use.

An example

Let’s solve the following linear programming problem using JuMP and HiGHS. We will first look at the complete code to solve the problem and then go through it step by step.

Here’s the problem:

And here’s the code to solve this problem:

julia> using JuMP


julia> using HiGHS


julia> model = Model(HiGHS.Optimizer)
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> @variable(model, x >= 0)
x

julia> @variable(model, 0 <= y <= 3)
y

julia> @objective(model, Min, 12x + 20y)
12 x + 20 y

julia> @constraint(model, c1, 6x + 8y >= 100)
c1 : 6 x + 8 y ≥ 100

julia> @constraint(model, c2, 7x + 12y >= 120)
c2 : 7 x + 12 y ≥ 120

julia> print(model)
Min 12 x + 20 y
Subject to
 c1 : 6 x + 8 y ≥ 100
 c2 : 7 x + 12 y ≥ 120
 x ≥ 0
 y ≥ 0
 y ≤ 3

julia> optimize!(model)
Running HiGHS 1.7.2 (git hash: 5ce7a2753): Copyright (c) 2024 HiGHS under MIT licence terms
Coefficient ranges:
  Matrix [6e+00, 1e+01]
  Cost   [1e+01, 2e+01]
  Bound  [3e+00, 3e+00]
  RHS    [1e+02, 1e+02]
Presolving model
2 rows, 2 cols, 4 nonzeros  0s
2 rows, 2 cols, 4 nonzeros  0s
Presolve : Reductions: rows 2(-0); columns 2(-0); elements 4(-0) - Not reduced
Problem not reduced by presolve: solving the LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0     0.0000000000e+00 Pr: 2(220) 0s
          2     2.0500000000e+02 Pr: 0(0) 0s
Model   status      : Optimal
Simplex   iterations: 2
Objective value     :  2.0500000000e+02
HiGHS run time      :          0.00

julia> termination_status(model)
OPTIMAL::TerminationStatusCode = 1

julia> primal_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

julia> dual_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

julia> objective_value(model)
204.99999999999997

julia> value(x)
15.000000000000005

julia> value(y)
1.249999999999996

julia> shadow_price(c1)
-0.24999999999999922

julia> shadow_price(c2)
-1.5000000000000007

Step-by-step

Once JuMP is installed, to use JuMP in your programs write:

julia> using JuMP

We also need to include a Julia package which provides an appropriate solver. We want to use HiGHS.Optimizer here which is provided by the HiGHS.jl package:

julia> using HiGHS

JuMP builds problems incrementally in a Model object. Create a model by passing an optimizer to the Model function:

julia> model = Model(HiGHS.Optimizer)
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

Variables are modeled using @variable:

julia> @variable(model, x >= 0)
x

The macro creates a new Julia object, x, in the current scope. We could have made this more explicit by writing:

x = @variable(model, x >= 0)

but the macro does this automatically for us to save writing x twice.

Variables can have lower and upper bounds:

julia> @variable(model, 0 <= y <= 30)
y

The objective is set using @objective:

julia> @objective(model, Min, 12x + 20y)
12 x + 20 y

Constraints are modeled using @constraint. Here, c1 and c2 are the names of our constraint:

julia> @constraint(model, c1, 6x + 8y >= 100)
c1 : 6 x + 8 y ≥ 100

julia> @constraint(model, c2, 7x + 12y >= 120)
c2 : 7 x + 12 y ≥ 120

Call print to display the model:

julia> print(model)
Min 12 x + 20 y
Subject to
 c1 : 6 x + 8 y ≥ 100
 c2 : 7 x + 12 y ≥ 120
 x ≥ 0
 y ≥ 0
 y ≤ 30

To solve the optimization problem, call the optimize! function:

julia> optimize!(model)
Running HiGHS 1.7.2 (git hash: 5ce7a2753): Copyright (c) 2024 HiGHS under MIT licence terms
Coefficient ranges:
  Matrix [6e+00, 1e+01]
  Cost   [1e+01, 2e+01]
  Bound  [3e+01, 3e+01]
  RHS    [1e+02, 1e+02]
Presolving model
2 rows, 2 cols, 4 nonzeros  0s
2 rows, 2 cols, 4 nonzeros  0s
Presolve : Reductions: rows 2(-0); columns 2(-0); elements 4(-0) - Not reduced
Problem not reduced by presolve: solving the LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0     0.0000000000e+00 Pr: 2(220) 0s
          2     2.0500000000e+02 Pr: 0(0) 0s
Model   status      : Optimal
Simplex   iterations: 2
Objective value     :  2.0500000000e+02
HiGHS run time      :          0.00

The ! after optimize is part of the name. It’s nothing special. Julia has a convention that functions which mutate their arguments should end in !. A common example is push!.

Now let’s see what information we can query about the solution, starting with is_solved_and_feasible:

julia> is_solved_and_feasible(model)
true

We can get more information about the solution by querying the three types of statuses.

termination_status tells us why the solver stopped:

julia> termination_status(model)
OPTIMAL::TerminationStatusCode = 1

In this case, the solver found an optimal solution.

Check primal_status to see if the solver found a primal feasible point:

julia> primal_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

and dual_status to see if the solver found a dual feasible point:

julia> dual_status(model)
FEASIBLE_POINT::ResultStatusCode = 1

Now we know that our solver found an optimal solution, and that it has a primal and a dual solution to query.

Query the objective value using objective_value:

julia> objective_value(model)
205.0

the primal solution using value:

julia> value(x)
15.000000000000004

julia> value(y)
1.2499999999999976

and the dual solution using shadow_price:

julia> shadow_price(c1)
-0.24999999999999917

julia> shadow_price(c2)
-1.5000000000000007

You should always check whether the solver found a solution before calling solution functions like value or objective_value. A common workflow is:

```julia
optimize!(model)
if !is_solved_and_feasible(model)
    error("Solver did not find an optimal solution")
end
```

That’s it for our simple model. In the rest of this tutorial, we expand on some of the basic JuMP operations.

Model basics

Create a model by passing an optimizer:

julia> model = Model(HiGHS.Optimizer)
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

Alternatively, call set_optimizer at any point before calling optimize!:

julia> model = Model()
A JuMP Model
├ solver: none
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> set_optimizer(model, HiGHS.Optimizer)

For some solvers, you can also use direct_model, which offers a more efficient connection to the underlying solver:

julia> model = direct_model(HiGHS.Optimizer())
A JuMP Model
├ mode: DIRECT
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

Some solvers do not support direct_model!

Solver Options

Pass options to solvers with optimizer_with_attributes:

julia> model =
           Model(optimizer_with_attributes(HiGHS.Optimizer, "output_flag" => false))
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

These options are solver-specific. To find out the various options available, see the GitHub README of the individual solver packages. The link to each solver’s GitHub page is in the Supported solvers table.

You can also pass options with set_attribute:

julia> model = Model(HiGHS.Optimizer)
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> set_attribute(model, "output_flag", false)

Solution basics

We saw above how to use termination_status and primal_status to understand the solution returned by the solver.

However, only query solution attributes like value and objective_value if there is an available solution. Here’s a recommended way to check:

julia> function solve_infeasible()
           model = Model(HiGHS.Optimizer)
           @variable(model, 0 <= x <= 1)
           @variable(model, 0 <= y <= 1)
           @constraint(model, x + y >= 3)
           @objective(model, Max, x + 2y)
           optimize!(model)
           if !is_solved_and_feasible(model)
               @warn("The model was not solved correctly.")
               return
           end
           return value(x), value(y)
       end
solve_infeasible (generic function with 1 method)

julia> solve_infeasible()
Running HiGHS 1.7.2 (git hash: 5ce7a2753): Copyright (c) 2024 HiGHS under MIT licence terms
Coefficient ranges:
  Matrix [1e+00, 1e+00]
  Cost   [1e+00, 2e+00]
  Bound  [1e+00, 1e+00]
  RHS    [3e+00, 3e+00]
Presolving model
Problem status detected on presolve: Infeasible
Model   status      : Infeasible
Objective value     :  0.0000000000e+00
HiGHS run time      :          0.00
ERROR:   No LP invertible representation for getDualRay
┌ Warning: The model was not solved correctly.
└ @ Main REPL[1]:9

Variable basics

Let’s create a new empty model to explain some of the variable syntax:

julia> model = Model()
A JuMP Model
├ solver: none
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

Variable bounds

All of the variables we have created till now have had a bound. We can also create a free variable.

julia> @variable(model, free_x)
free_x

While creating a variable, instead of using the <= and >= syntax, we can also use the lower_bound and upper_bound keyword arguments.

julia> @variable(model, keyword_x, lower_bound = 1, upper_bound = 2)
keyword_x

We can query whether a variable has a bound using the has_lower_bound and has_upper_bound functions. The values of the bound can be obtained using the lower_bound and upper_bound functions.

julia> has_upper_bound(keyword_x)
true

julia> upper_bound(keyword_x)
2.0

Note querying the value of a bound that does not exist will result in an error.

julia>     lower_bound(free_x)
Variable free_x does not have a lower bound.

Containers

We have already seen how to add a single variable to a model using the @variable macro. Now let’s look at ways to add multiple variables to a model.

JuMP provides data structures for adding collections of variables to a model. These data structures are referred to as containers and are of three types: Arrays, DenseAxisArrays, and SparseAxisArrays.

Arrays

JuMP arrays are created when you have integer indices that start at 1:

julia> @variable(model, a[1:2, 1:2])
2×2 Matrix{VariableRef}:
 a[1,1]  a[1,2]
 a[2,1]  a[2,2]

Index elements in a as follows:

julia> a[1, 1]
a[1,1]

julia> a[2, :]
2-element Vector{VariableRef}:
 a[2,1]
 a[2,2]

Create an n-dimensional variable with bounds ( ) as follows:

julia> n = 10
10

julia> l = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10];


julia> u = [10, 11, 12, 13, 14, 15, 16, 17, 18, 19];


julia> @variable(model, l[i] <= x[i = 1:n] <= u[i])
10-element Vector{VariableRef}:
 x[1]
 x[2]
 x[3]
 x[4]
 x[5]
 x[6]
 x[7]
 x[8]
 x[9]
 x[10]

We can also create variable bounds that depend upon the indices:

julia> @variable(model, y[i = 1:2, j = 1:2] >= 2i + j)
2×2 Matrix{VariableRef}:
 y[1,1]  y[1,2]
 y[2,1]  y[2,2]

DenseAxisArrays

DenseAxisArrays are used when the indices are not one-based integer ranges. The syntax is similar except with an arbitrary vector as an index as opposed to a one-based range:

julia> @variable(model, z[i = 2:3, j = 1:2:3] >= 0)
2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
    Dimension 1, 2:3
    Dimension 2, 1:2:3
And data, a 2×2 Matrix{VariableRef}:
 z[2,1]  z[2,3]
 z[3,1]  z[3,3]

Indices do not have to be integers. They can be any Julia type:

julia> @variable(model, w[1:5, ["red", "blue"]] <= 1)
2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
    Dimension 1, Base.OneTo(5)
    Dimension 2, ["red", "blue"]
And data, a 5×2 Matrix{VariableRef}:
 w[1,red]  w[1,blue]
 w[2,red]  w[2,blue]
 w[3,red]  w[3,blue]
 w[4,red]  w[4,blue]
 w[5,red]  w[5,blue]

Index elements in a DenseAxisArray as follows:

julia> z[2, 1]
z[2,1]

julia> w[2:3, ["red", "blue"]]
2-dimensional DenseAxisArray{VariableRef,2,...} with index sets:
    Dimension 1, [2, 3]
    Dimension 2, ["red", "blue"]
And data, a 2×2 Matrix{VariableRef}:
 w[2,red]  w[2,blue]
 w[3,red]  w[3,blue]

See Forcing the container type for more details.

SparseAxisArrays

SparseAxisArrays are created when the indices do not form a Cartesian product. For example, this applies when indices have a dependence upon previous indices (called triangular indexing):

julia> @variable(model, u[i = 1:2, j = i:3])
SparseAxisArray{VariableRef, 2, Tuple{Int64, Int64}} with 5 entries:
  [1, 1]  =  u[1,1]
  [1, 2]  =  u[1,2]
  [1, 3]  =  u[1,3]
  [2, 2]  =  u[2,2]
  [2, 3]  =  u[2,3]

We can also conditionally create variables by adding a comparison check that depends upon the named indices and is separated from the indices by a semi-colon ;:

julia> @variable(model, v[i = 1:9; mod(i, 3) == 0])
SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 3 entries:
  [3]  =  v[3]
  [6]  =  v[6]
  [9]  =  v[9]

Index elements in a DenseAxisArray as follows:

julia> u[1, 2]
u[1,2]

julia> v[[3, 6]]
SparseAxisArray{VariableRef, 1, Tuple{Int64}} with 2 entries:
  [3]  =  v[3]
  [6]  =  v[6]

Integrality

JuMP can create binary and integer variables. Binary variables are constrained to the set , and integer variables are constrained to the set .

Integer variables

Create an integer variable by passing Int:

julia> @variable(model, integer_x, Int)
integer_x

or setting the integer keyword to true:

julia> @variable(model, integer_z, integer = true)
integer_z

Binary variables

Create a binary variable by passing Bin:

julia> @variable(model, binary_x, Bin)
binary_x

or setting the binary keyword to true:

julia> @variable(model, binary_z, binary = true)
binary_z

Constraint basics

We’ll need a new model to explain some of the constraint basics:

julia> model = Model()
A JuMP Model
├ solver: none
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> @variable(model, x)
x

julia> @variable(model, y)
y

julia> @variable(model, z[1:10]);

Containers

Just as we had containers for variables, JuMP also provides Arrays, DenseAxisArrays, and SparseAxisArrays for storing collections of constraints. Examples for each container type are given below.

Arrays

Create an Array of constraints:

julia> @constraint(model, [i = 1:3], i * x <= i + 1)
3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
 x ≤ 2
 2 x ≤ 3
 3 x ≤ 4

DenseAxisArrays

Create an DenseAxisArray of constraints:

julia> @constraint(model, [i = 1:2, j = 2:3], i * x <= j + 1)
2-dimensional DenseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape},2,...} with index sets:
    Dimension 1, Base.OneTo(2)
    Dimension 2, 2:3
And data, a 2×2 Matrix{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:
 x ≤ 3    x ≤ 4
 2 x ≤ 3  2 x ≤ 4

SparseAxisArrays

Create an SparseAxisArray of constraints:

julia> @constraint(model, [i = 1:2, j = 1:2; i != j], i * x <= j + 1)
SparseAxisArray{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}, 2, Tuple{Int64, Int64}} with 2 entries:
  [1, 2]  =  x ≤ 3
  [2, 1]  =  2 x ≤ 2

Constraints in a loop

We can add constraints using regular Julia loops:

julia> for i in 1:3
           @constraint(model, 6x + 4y >= 5i)
       end

or use for each loops inside the @constraint macro:

julia> @constraint(model, [i in 1:3], 6x + 4y >= 5i)
3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:
 6 x + 4 y ≥ 5
 6 x + 4 y ≥ 10
 6 x + 4 y ≥ 15

We can also create constraints such as :

julia> @constraint(model, sum(z[i] for i in 1:10) <= 1)
z[1] + z[2] + z[3] + z[4] + z[5] + z[6] + z[7] + z[8] + z[9] + z[10] ≤ 1

Objective functions

Set an objective function with @objective:

julia> model = Model(HiGHS.Optimizer)
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> @variable(model, x >= 0)
x

julia> @variable(model, y >= 0)
y

julia> @objective(model, Min, 2x + y)
2 x + y

Create a maximization objective using Max:

julia> @objective(model, Max, 2x + y)
2 x + y

Calling @objective multiple times will over-write the previous objective. This can be useful when you want to solve the same problem with different objectives.

Vectorized syntax

We can also add constraints and an objective to JuMP using vectorized linear algebra. We’ll illustrate this by solving an LP in standard form that is,

julia> vector_model = Model(HiGHS.Optimizer)
A JuMP Model
├ solver: HiGHS
├ objective_sense: FEASIBILITY_SENSE
├ num_variables: 0
├ num_constraints: 0
└ Names registered in the model: none

julia> A = [1 1 9 5; 3 5 0 8; 2 0 6 13]
3×4 Matrix{Int64}:
 1  1  9   5
 3  5  0   8
 2  0  6  13

julia> b = [7, 3, 5]
3-element Vector{Int64}:
 7
 3
 5

julia> c = [1, 3, 5, 2]
4-element Vector{Int64}:
 1
 3
 5
 2

julia> @variable(vector_model, x[1:4] >= 0)
4-element Vector{VariableRef}:
 x[1]
 x[2]
 x[3]
 x[4]

julia> @constraint(vector_model, A * x .== b)
3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.EqualTo{Float64}}, ScalarShape}}:
 x[1] + x[2] + 9 x[3] + 5 x[4] = 7
 3 x[1] + 5 x[2] + 8 x[4] = 3
 2 x[1] + 6 x[3] + 13 x[4] = 5

julia> @objective(vector_model, Min, c' * x)
x[1] + 3 x[2] + 5 x[3] + 2 x[4]

julia> optimize!(vector_model)
Running HiGHS 1.7.2 (git hash: 5ce7a2753): Copyright (c) 2024 HiGHS under MIT licence terms
Coefficient ranges:
  Matrix [1e+00, 1e+01]
  Cost   [1e+00, 5e+00]
  Bound  [0e+00, 0e+00]
  RHS    [3e+00, 7e+00]
Presolving model
3 rows, 4 cols, 10 nonzeros  0s
3 rows, 4 cols, 10 nonzeros  0s
Presolve : Reductions: rows 3(-0); columns 4(-0); elements 10(-0) - Not reduced
Problem not reduced by presolve: solving the LP
Using EKK dual simplex solver - serial
  Iteration        Objective     Infeasibilities num(sum)
          0     0.0000000000e+00 Pr: 3(13.5) 0s
          4     4.9230769231e+00 Pr: 0(0) 0s
Model   status      : Optimal
Simplex   iterations: 4
Objective value     :  4.9230769231e+00
HiGHS run time      :          0.00

julia> @assert is_solved_and_feasible(vector_model)


julia> objective_value(vector_model)
4.923076923076922