Quick Start Guide

Quick Start Guide

This quick start guide will introduce the main concepts of JuMP. If you are familiar with another modeling language embedded in a high-level language such as PuLP (Python) or a solver-specific interface you will find most of this familiar. If you are coming from an AMPL or similar background, you may find some of the concepts novel but the general appearance will still be familiar.

The example in this guide is deliberately kept simple. There are more complex examples in the JuMP/examples/ folder.

Once JuMP is installed, to use JuMP in your programs, you just need to say:

julia> using JuMP

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

julia> using GLPK

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

Models are created with the Model() function. The with_optimizer syntax is used to specify the optimizer to be used:

julia> model = Model(with_optimizer(GLPK.Optimizer))
A JuMP Model
Note

Your model doesn't have to be called model - it's just a name.

There are a few options for defining a variable, depending on whether you want to have lower bounds, upper bounds, both bounds, or even no bounds. The following commands will create two variables, x and y, with both lower and upper bounds. Note the first argument is our model variable $model$. These variables are associated with this model and cannot be used in another model.

julia> @variable(model, 0 <= x <= 2)
x

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

See the Variables section for more information on creating variables.

Next we'll set our objective. Note again the model, so we know which model's objective we are setting! The objective sense, Max or Min, should be provided as the second argument. Note also that we don't have a multiplication * symbol between 5 and our variable x - Julia is smart enough to not need it! Feel free to stick with * if it makes you feel more comfortable, as we have done with 3 * y. (We have been intentionally inconsistent here to demonstrate different syntax; however, it is good practice to pick one way or the other consistently in your code.)

julia> @objective(model, Max, 5x + 3 * y)

Adding constraints is a lot like setting the objective. Here we create a less-than-or-equal-to constraint using <=, but we can also create equality constraints using == and greater-than-or-equal-to constraints with >=:

julia> @constraint(model, con, 1x + 5y <= 3)
con : x + 5 y <= 3.0

Note that in a similar manner to the @variable macro, we have named the constraint con. This will bind the constraint to the Julia variable con for later analysis.

Models are solved with the JuMP.optimize! function:

julia> JuMP.optimize!(model)

After the call to JuMP.optimize! has finished, we need to understand why the optimizer stopped. This can be for a number of reasons. First, the solver might have found the optimal solution, or proved that the problem is infeasible. However, it might also have run into numerical difficulties, or terminated due to a setting such as a time limit. We can ask the solver why it stopped using the JuMP.termination_status function:

julia> JuMP.termination_status(model)
Success::TerminationStatusCode = 0

In this case, GLPK returned Success. This does not mean that it has found the optimal solution. Instead, it indicates that GLPK has finished running and did not encounter any errors or user-provided termination limits.

To understand the reason for termination in more detail, we need to query JuMP.primalstatus:

julia> JuMP.primal_status(model)
FeasiblePoint::ResultStatusCode = 1

This indicates that GLPK has found a FeasiblePoint to the primal problem. Coupled with the Success from JuMP.termination_status, we can infer that GLPK has indeed found the optimal solution. We can also query JuMP.dual_status:

julia> JuMP.dual_status(model)
FeasiblePoint::ResultStatusCode = 1

Like the primal_status, GLPK indicates that it has found a FeasiblePoint to the dual problem.

Finally, we can query the result of the optimization. First, we can query the objective value:

julia> JuMP.objective_value(model)
10.6

We can also query the primal result values of the x and y variables:

julia> JuMP.result_value(x)
2.0

julia> JuMP.result_value(y)
0.2

We can also query the value of the dual variable associated with the constraint con (which we bound to a Julia variable when defining the constraint):

julia> JuMP.result_dual(con)
-0.6

To query the dual variables associated with the variable bounds, things are a little trickier as we first need to obtain a reference to the constraint:

julia> x_upper = JuMP.UpperBoundRef(x)
x <= 2.0

julia> JuMP.result_dual(x_upper)
-4.4

A similar process can be followed to obtain the dual of the lower bound constraint on y:

julia> y_lower = JuMP.LowerBoundRef(y)
y >= 0.0

julia> JuMP.result_dual(y_lower)
0.0