Gurobi.jl

The Gurobi Optimizer is a commercial optimization solver for a variety of mathematical programming problems, including linear programming (LP), quadratic programming (QP), quadratically constrained programming (QCP), mixed integer linear programming (MILP), mixed-integer quadratic programming (MIQP), and mixed-integer quadratically constrained programming (MIQCP).

The Gurobi wrapper for Julia is community driven and not officially supported by Gurobi. If you are a commercial customer interested in official support for Julia from Gurobi, let them know!

Using Gurobi v0.9 and you got an error like `Q not PSD`?

You need to set the NonConvex parameter:

model = Model(with_optimizer(Gurobi.Optimizer, NonConvex = 2))

Use with JuMP

We highly recommend that you use the Gurobi.jl package with higher level packages such as JuMP.jl.

This can be done using the Gurobi.Optimizer object. Here is how to create a JuMP model that uses Gurobi as the solver. Parameters are passed as keyword arguments:

using JuMP, Gurobi

model = Model(with_optimizer(Gurobi.Optimizer, Presolve=0, OutputFlag=0))

See the Gurobi Documentation for a list and description of allowable parameters.

Common Performance Pitfall with JuMP

Gurobi API works differently than most solvers. Any changes to the model are not applied immediately, but instead go sit in a internal buffer (making any modifications appear to be instantaneous) waiting for a call to update_model! (where the work is done). If Gurobi.jl is used directly, it is the user responsability to call update_model! when necessary (for example, before solving the model), as it would be if the user was using the official C interface. However, if Gurobi.jl is used with JuMP, it becomes Gurobi.jl responsibility to call update_model! when necessary, as a valid JuMP program should work for solvers with and without such lazy update semantics (i.e., with and without a update_model!-like method).

This leads to a common performance pitfall that has the following message as its main symptom: Warning: excessive time spent in model updates. Consider calling update less frequently. This often means the JuMP program was structured in such a way that Gurobi.jl ends up calling update_model! each iteration of a loop. Usually, it is possible (and easy) to restructure the JuMP program in a way it stays solver-agnostic and has a close-to-ideal performance with Gurobi. To guide such restructuring it is good to keep in mind the following bits of information:

update_model! is only called if changes were done since last update_model! (i.e., the internal buffer is not empty).
update_model! is called when JuMP.optimize! is called, but this often is not the source of the problem.
update_model!may be called when ANY model attribute is queried even if that specific attribute was not changed, and this often the source of the problem.
The worst-case scenario is, therefore, a loop of modify-query-modify-query, even if what is being modified and what is being queried are two completely distinct things.

Finally, for an example, prefer:

# GOOD
model = Model(Gurobi.Optimizer)
@variable(model, x[1:100] >= 0)
for i = 1:100  # all modifications are done before any queries
    set_upper_bound(x[i], i)
end
for i = 1:100 # only the first `lower_bound` query may trigger an `update_model!` 
    println(lower_bound(x[i]))
end

to:

# BAD
model = Model(Gurobi.Optimizer)
@variable(model, x[1:100] >= 0)
for i = 1:100
    set_upper_bound(x[i], i)
    # there is a potential `update_model!` each iteration of this loop
    println(lower_bound(x[i]))
end

Installation

Here is the procedure to setup this package:

Obtain a license of Gurobi and install Gurobi solver, following the instructions on Gurobi's website.
The minimum version supported by Gurobi.jl is Gurobi v7.0.
Make sure the GUROBI_HOME environmental variable is set to the path of the Gurobi directory. This is part of a standard installation. The Gurobi library will be searched for in GUROBI_HOME/lib on unix platforms and GUROBI_HOME\bin on Windows. If the library is not found, check that your version is listed in deps/build.jl.
Install this package using Pkg.add("Gurobi").
Now, you can start using it.

By default, building Gurobi.jl will fail if the Gurobi library is not found. This may not be desirable in certain cases, for example when part of a package's test suite uses Gurobi as an optional test dependency, but Gurobi cannot be installed on a CI server running the test suite. To support this use case, the GUROBI_JL_SKIP_LIB_CHECK environment variable may be set (to any value) to make Gurobi.jl installable (but not usable).

"LoadError: Unable to locate Gurobi installation"

Make sure that you have downloaded and installed Gurobi from gurobi.com. Also, make sure that you validate the license.
Check that your Gurobi version is at least 7.0
Make sure that the GUROBI_HOME environment variable is set correctly. You can see the current value as follows

julia> ENV["GUROBI_HOME"]
"C:\\gurobi801\\win64"

If it is not set correctly (e.g., you get an error Key "GUROBI_HOME" not found), you can set it as follows

julia> ENV["GUROBI_HOME"] = "/replace/this/with/the/path/to/gurobi"

julia> import Pkg; Pkg.build("Gurobi")

The Gurobi library (gurobiXXX.dll on Windows, gurobiXXX.so on Unix, and gurobiXXX.dylib in OSX where XXX is a version) will be searched for in GUROBI_HOME/lib on unix platforms and GUROBI_HOME\bin on Windows.

Reusing the same Gurobi environment for multiple solves

When using this package via other packages such as JuMP.jl, the default behavior is to obtain a new Gurobi license token every time a model is created and solved. If you are using Gurobi in a setting where the number of concurrent Gurobi uses is limited (e.g. "Single-Use" or "Floating-Use" licenses), you might instead prefer to obtain a single license token that is shared by all models that your program solves. You can do this by passing a Gurobi Environment object as the first parameter to Gurobi.Optimizer. For example, the follow code snippet solves multiple problems with JuMP using the same license token:

using JuMP, Gurobi

const GRB_ENV = Gurobi.Env()

model1 = Model(with_optimizer(Gurobi.Optimizer, GRB_ENV))
...

# The solvers can have different options too
model2 = Model(with_optimizer(Gurobi.Optimizer, GRB_ENV, OutputFlag=0))
...

Accessing Gurobi-specific attributes via JuMP

You can get and set Gurobi-specific variable, constraint, and model attributes via JuMP as follows:

using JuMP, Gurobi

model = JuMP.direct_model(Gurobi.Optimizer(OutputFlag=0))
@variable(model, x >= 0)
@constraint(model, c, 2x >= 1)
@objective(model, Min, x)
MOI.set(model, Gurobi.ConstraintAttribute("Lazy"), c, 2)
optimize!(model)

MOI.get(model, Gurobi.VariableAttribute("LB"), x)  # Returns 0.0

MOI.get(model, Gurobi.ModelAttribute("NumConstrs")) # Returns 1

Note that we are using JuMP in direct-mode. A complete list of supported Gurobi attributes can be found in their online documentation.

Most users should not need to use the low-level API detailed in the following sections.

API Overview

This package provides both APIs at different levels for constructing models and solving optimization problems.

Gurobi Environment

A Gurobi model is always associated with an Gurobi environment, which maintains a solver configuration. By setting parameters to this environment, one can control or tune the behavior of a Gurobi solver.

To construct a Gurobi Environment, one can write:

env = Gurobi.Env()

This package provides functions to get and set parameters:

getparam(env, name)       # get the value of a parameter
setparam!(env, name, v)   # set the value of a parameter
setparams!(env, name1=value1, name2=value2, ...)  # set parameters using keyword arguments

You may refer to Gurobi's Parameter Reference for the whole list of parameters.

Here are some simple examples

setparam!(env, "Method", 2)   # choose to use Barrier method
setparams!(env; IterationLimit=100, Method=1) # set the maximum iterations and choose to use Simplex method

These parameters may be used directly with the GurobiSolver object used by MathProgBase. For example:

solver = GurobiSolver(Method=2)
solver = GurobiSolver(Method=1, IterationLimit=100.)

High-level API

If the objective coefficients and the constraints have already been given, one may use a high-level function gurobi_model to construct a model:

gurobi_model(env, ...)

One can use keyword arguments to specify the models:

name: the model name.
sense: the sense of optimization (a symbol, which can be either :minimize (default) or :maximize).
f: the linear coefficient vector.
H: the quadratic coefficient matrix (can be dense or sparse).
A: the coefficient matrix of the linear inequality constraints.
b: the right-hand-side of the linear inequality constraints.
Aeq: the coefficient matrix of the equality constraints.
beq: the right-hand-side of the equality constraints.
lb: the variable lower bounds.
ub: the variable upper bounds.

This function constructs a model that represents the following problem:

objective:  (1/2) x' H x + f' x

      s.t.   A x <= b
           Aeq x = beq
         lb <= x <= ub

The caller must specify f using a non-empty vector, while other keyword arguments are optional. When H is omitted, this reduces to an LP problem. When lb is omitted, the variables are not lower bounded, and when ub is omitted, the variables are not upper bounded.

Low-level API

This package also provides functions to build the model from scratch and gradually add variables and constraints. To construct an empty model, one can write:

env = Gurobi.Env()    # creates a Gurobi environment

model = Gurobi.Model(env, name)   # creates an empty model
model = Gurobi.Model(env, name, sense)

Here, sense is a symbol, which can be either :minimize or :maximize (default to :minimize when omitted).

Then, the following functions can be used to add variables and constraints to the model:

## add variables

add_var!(model, vtype, c)   # add an variable with coefficient c
                            # vtype can be either of
                            # - GRB_CONTINUOUS  (for continuous variable)
                            # - GRB_INTEGER (for integer variable)
                            # - GRB_BINARY (for binary variable, i.e. 0/1)

add_cvar!(model, c)            # add a continuous variable
add_cvar!(model, c, lb, ub)    # add a continuous variable with specified bounds

add_ivar!(model, c)            # add an integer variable
add_ivar!(model, c, lb, ub)    # add an integer variable with specified bounds

add_bvar!(model, c)            # add a binary variable

## add constraints

# add a constraint with non-zero coefficients on specific variables.
# rel can be '<', '>', or '='
add_constr!(model, inds, coeffs, rel, rhs)

# add a constraint with coefficient vector for all variables.
add_constr!(model, coeffs, rel, rhs)

# add constraints using CSR format
add_constrs!(model, cbegin, inds, coeffs, rel, rhs)

# add constraints using a matrix: A x (rel) rhs
add_constrs!(model, A, rel, rhs)  # here A can be dense or sparse

# add constraints using a transposed matrix: At' x (rel) rhs
# this is usually more efficient than add_constrs!
add_constrs_t!(model, At, rel, rhs)  # here At can be dense or sparse

# add a range constraint
add_rangeconstr!(model, inds, coeffs, lb, ub)

# add range constraints using CSR format
add_rangeconstrs!(model, cbegin, inds, coeffs, lb, ub)

# add range constraints using a matrix:  lb <= A x <= ub
add_rangeconstrs!(model, A, lb, ub)  # here A can be dense or sparse

# add range constraints using a transposed matrix: lb <= At' x <= ub
# this is usually more efficient than add_rangeconstrs!
add_rangeconstrs_t!(model, At, lb, ub)  # here At can be dense or sparse

Modify Problem

It is not uncommon in practice that one would like to adjust the objective coefficients and solve the problem again. This package provides a function set_objcoeffs! for this purpose:

set_objcoeffs!(model, new_coeffs)
 # ... one can also call add_constr! and friends to add additional constraints ...
update_model!(model)   # changes take effect after this
optimize(model)

Examples

The usage of this package is straight forward. Below, we use several examples to demonstrate the use of this package to solve optimization problems.

Linear Programming Examples

Problem formulation:

maximize x + y

s.t. 50 x + 24 y <= 2400
     30 x + 33 y <= 2100
     x >= 45, y >= 5

Below, we show how this problem can be constructed and solved in different ways.

Example 1.1: High-level Linear Programming API

Using the gurobi_model function:

using Gurobi

env = Gurobi.Env()

# set presolve to 0
setparam!(env, "Presolve", 0)

 # construct the model
model = gurobi_model(env;
    name = "lp_01",
    f = ones(2),
    A = [50. 24.; 30. 33.],
    b = [2400., 2100.],
    lb = [5., 45.])

 # run optimization
optimize(model)

 # show results
sol = get_solution(model)
println("soln = $(sol)")

objv = get_objval(model)
println("objv = $(objv)")

Example 1.2: Low-level Linear Programming API

using Gurobi

env = Gurobi.Env()

# set presolve to 0
setparam!(env, "Presolve", 0)

 # creates an empty model ("lp_01" is the model name)
model = Gurobi.Model(env, "lp_01", :maximize)

 # add variables
 # add_cvar!(model, obj_coef, lower_bound, upper_bound)
add_cvar!(model, 1.0, 45., Inf)  # x: x >= 45
add_cvar!(model, 1.0,  5., Inf)  # y: y >= 5

 # For Gurobi, you have to call update_model to have the
 # lastest changes take effect
update_model!(model)

 # add constraints
 # add_constr!(model, coefs, sense, rhs)
add_constr!(model, [50., 24.], '<', 2400.) # 50 x + 24 y <= 2400
add_constr!(model, [30., 33.], '<', 2100.) # 30 x + 33 y <= 2100
update_model!(model)

println(model)

 # perform optimization
optimize(model)

You may also add variables and constraints in batch, as:

 # add mutliple variables in batch
add_cvars!(model, [1., 1.], [45., 5.], Inf)

 # add multiple constraints in batch
A = [50. 24.; 30. 33.]
b = [2400., 2100.]
add_constrs!(model, A, '<', b)

Example 1.3: Linear programming (MATLAB-like style)

You may also specify and solve the entire problem in one function call, using the solver-independent MathProgBase package.

Julia code:

using MathProgBase, Gurobi

f = [1., 1.]
A = [50. 24.; 30. 33.]
b = [2400., 2100.]
lb = [5., 45.]

# pass params as keyword arguments to GurobiSolver
solution = linprog(f, A, '<', b, lb, Inf, GurobiSolver(Presolve=0))

Example 1.4: Linear programming with JuMP (Algebraic model)

Using JuMP, we can specify linear programming problems using a more natural algebraic approach.

using JuMP, Gurobi

# pass params as keyword arguments to GurobiSolver
model = Model(with_optimizer(Gurobi.Optimizer, Presolve=0))

@variable(model, x >= 5)
@variable(model, y >= 45)

@objective(model, Min, x + y)
@constraint(model, 50x + 24y <= 2400)
@constraint(model, 30x + 33y <= 2100)

optimize!(model)
println("Optimal objective: ", objective_value(model),
	". x = ", value(x), " y = ", value(y))

Quadratic programming Examples

Problem formulation:

minimize x^2 + xy + y^2 + yz + z^2

s.t.  x + 2 y + 3 z >= 4
      x +   y       >= 1

Example 2.1: High-level Quadratic Programming API

using the function gurobi_model:

using Gurobi

env = Gurobi.Env()

model = gurobi_model(env;
        name = "qp_01",
        H = [2. 1. 0.; 1. 2. 1.; 0. 1. 2.],
        f = [0., 0., 0.],
        A = -[1. 2. 3.; 1. 1. 0.],
        b = -[4., 1.])
optimize(model)

Example 2.2: Low-level Quadratic Programming API

using Gurobi

env = Gurobi.Env()

model = Gurobi.Model(env, "qp_01")

add_cvars!(model, [1., 1.], 0., Inf)
update_model!(model)

 # add quadratic terms: x^2, x * y, y^2
 # add_qpterms!(model, rowinds, colinds, coeffs)
add_qpterms!(model, [1, 1, 2], [1, 2, 2], [1., 1., 1.])

 # add linear constraints
add_constr!(model, [1., 2., 3.], '>', 4.)
add_constr!(model, [1., 1., 0.], '>', 1.)
update_model!(model)

optimize(model)

Mixed Integer Programming

This package also supports mixed integer programming.

Problem formulation:

maximize x + 2 y + 5 z

s.t.  x + y + z <= 10
      x + 2 y + z <= 15
      x is continuous: 0 <= x <= 5
      y is integer: 0 <= y <= 10
      z is binary

Example 3.1: Low-level MIP API

Julia code:

using Gurobi

env = Gurobi.Env()

model = Gurobi.Model(env, "mip_01", :maximize)

 # add continuous variable
add_cvar!(model, 1., 0., 5.)  # x

 # add integer variable
add_ivar!(model, 2., 0, 10)   # y

 # add binary variable
add_bvar!(model, 5.)          # z

 # have the variables incorporated into the model
update_model!(model)

add_constr!(model, ones(3), '<', 10.)
add_constr!(model, [1., 2., 1.], '<', 15.)

optimize(model)

Note that you can use add_ivars! and add_bvars! to add multiple integer or binary variables in batch.

Example 3.2: MIP using JuMP with Gurobi

using JuMP, Gurobi

model = Model(with_optimizer(Gurobi.Optimizer))

@variables(model, begin
    0 <= x <= 5
    0 <= y <= 10, Int
    z, Bin
end)

@objective(model, Max, x + 2y + 5z)
@constraint(model, x + y + z <= 10)
@constraint(model, x + 2y + z <= 15)

optimize!(model)

Quadratic constraints

The add_qconstr! function may be used to add quadratic constraints to a model.

Problem formulation:

maximize x + y

s.t.  x, y >= 0
      x^2 + y^2 <= 1

Julia code:

using Gurobi

env = Gurobi.Env()

model = Gurobi.Model(env, "qcqp_01", :maximize)

add_cvars!(model, [1., 1.], 0., Inf)
update_model!(model)

 # add_qpconstr!(model, linearindices, linearcoeffs, qrowinds, qcolinds, qcoeffs, sense, rhs)
add_qconstr!(model, [], [], [1, 2], [1, 2], [1, 1.], '<', 1.0)
update_model!(model)

optimize(model)

SOCP constraints of the form x'x <= y^2 and x'x <= yz can be added using this method as well.

Gurobi.jl

Using Gurobi v0.9 and you got an error like Q not PSD?

Use with JuMP

Common Performance Pitfall with JuMP

Installation

"LoadError: Unable to locate Gurobi installation"

Reusing the same Gurobi environment for multiple solves

Accessing Gurobi-specific attributes via JuMP

API Overview

Gurobi Environment

High-level API

Low-level API

Modify Problem

Examples

Linear Programming Examples

Example 1.1: High-level Linear Programming API

Example 1.2: Low-level Linear Programming API

Example 1.3: Linear programming (MATLAB-like style)

Example 1.4: Linear programming with JuMP (Algebraic model)

Quadratic programming Examples

Example 2.1: High-level Quadratic Programming API

Example 2.2: Low-level Quadratic Programming API

Mixed Integer Programming

Example 3.1: Low-level MIP API

Example 3.2: MIP using JuMP with Gurobi

Quadratic constraints

Using Gurobi v0.9 and you got an error like `Q not PSD`?