# 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
Feasibility problem with:
Variables: 0
Model mode: AUTOMATIC
CachingOptimizer state: NO_OPTIMIZER
Solver name: No optimizer attached.
```

The term "solver" is used as a synonym for "optimizer". The convention in code, however, is to always use "optimizer", e.g., `with_optimizer`

and `GLPK.Optimizer`

.

Your model doesn't have to be called `model`

- it's just a name.

The following commands will create two variables, `x`

and `y`

, with both lower and upper bounds. Note the first argument is our model `model`

. These variables (`x`

and `y`

) 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, including the syntax for specifying different combinations of bounds, i.e., only lower bounds, only upper bounds, or no bounds.

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 use `*`

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)
5 x + 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> optimize!(model)`

After the call to `JuMP.optimize!`

has finished, we need to query what happened. The solve could terminate 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> termination_status(model)
OPTIMAL::TerminationStatusCode = 1
```

In this case, `GLPK`

returned `OPTIMAL`

, this mean that it has found the optimal solution.

As the solver found an optimal solution, we expect the solution returned to be a primal-dual pair of feasible solutions with zero duality gap. We can verify the primal and dual status as follows to confirm this:

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

Note that the primal and dual status only inform that the primal and dual solutions are feasible and it is only because we verified that the termination status is `OPTIMAL`

that we can conclude that they form an optimal solution.

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

```
julia> objective_value(model)
10.6
```

We can also query the primal result values of the `x`

and `y`

variables:

```
julia> value(x)
2.0
julia> 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> dual(con)
-0.6
```

See the duality section for a discussion of the convention that JuMP uses for signs of duals.

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 = UpperBoundRef(x)
x <= 2.0
julia> 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 = LowerBoundRef(y)
y >= 0.0
julia> dual(y_lower)
0.0
```