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 solver is considered one of the best solvers (in terms of performance and success rate of tackling hard problems) in math programming, and its performance is comparable to (and sometimes superior to) CPLEX. Academic users can obtain a Gurobi license for free.

This package is a wrapper of the Gurobi solver (through its C interface). Currently, this package supports the following types of problems:

• Linear programming (LP)
• Mixed Integer Linear Programming (MILP)
• Quadratic programming (QP)
• Mixed Integer Quadratic Programming (MIQP)
• Quadratically constrained quadratic programming (QCQP)
• Second order cone programming (SOCP)
• Mixed integer second order cone programming (MISOCP)

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!

Installation

Here is the procedure to setup this package:

1. Obtain a license of Gurobi and install Gurobi solver, following the instructions on Gurobi's website.

2. Install this package using Pkg.add("Gurobi").

3. 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.

4. Now, you can start using it.

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

Use Other Packages

The Gurobi.jl package also works with other packages, including MathProgBase.jl and JuMP.jl, as a backend provider.

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. In all examples below, we assume that the preamble codes like the following exist in the script:

using Gurobi
env = Gurobi.Env()
... optional codes to set parameters to env ...

Example 1.1: High-level Linear Programming API

Using the gurobi_model function:

 # 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

 # 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

f = [1., 1.]
A = [50. 24.; 30. 33.]
b = [2400., 2100.]
lb = [5., 45.]
 
solution = linprog(f, A, '<', b, lb, Inf, GurobiSolver())

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

m = Model(solver=GurobiSolver())

@defVar(m, x >= 5)
@defVar(m, y >= 45)

@setObjective(m, Min, x + y)
@addConstraint(m, 50x + 24y <= 2400)
@addConstraint(m, 30x + 33y <= 2100)

status = solve(m)
println("Optimal objective: ",getObjectiveValue(m), 
	". x = ", getValue(x), " y = ", getValue(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:

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

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:

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

m = Model(solver=GurobiSolver())

@defVar(m, 0 <= x <= 5)
@defVar(m, 0 <= y <= 10, Int)
@defVar(m, z, Bin)

@setObjective(m, Max, x + 2y + 5z)
@addConstraint(m, x + y + z <= 10)
@addConstraint(m, x + 2y + z <= 15)

solve(m)

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:

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.