# Solving models with the Gurobi R interface

##

Solving models with the Gurobi R interface

The Gurobi R interface can be used to solve optimization problems of
the following form:

minimize | ||

subject to | (linear constraints) | |

(bound constraints) | ||

some integral | (integrality constraints) | |

some lie within second order cones | (cone constraints) | |

(quadratic constraints) | ||

some in SOS | (special ordered set constraints) |

Many of the model components listed here are optional. For example, integrality constraints may be omitted. We'll discuss the details of how models are represented shortly.

The following function allows you to take a model represented using R data structures and solve it with the Gurobi Optimizer:

gurobi | ( model, params=NULL ) |

The two arguments to this function are R `list`

variables, each
consisting of multiple *named components*. The first argument
contains the optimization model to be solved. The second contains an
optional list of Gurobi parameters to be modified during the solution
process. The return value of this function is a list, also consisting
of multiple named components. It contains the result of performing
the optimization on the specified model. We'll now discuss the
details of each of these lists.

The optimization model

As we've mentioned, the `model`

argument to the `gurobi()`

function is a `list`

variable, containing multiple
*named components* that represent the various parts
of the optimization
model. Several of these components are optional.
Note that you refer to a
named component of an R list variable by appending a dollar sign
followed by the component name to the list variable name. For
example, `model$A`

refers to component `A`

of list
variable `model`

.

The following is an enumeration of all of the named components of the
`model`

argument that Gurobi will take into account when
optimizing the model:

**A**- The linear constraint matrix. This can be dense or sparse.
Sparse matrices should be built using either
`sparseMatrix`

from the`Matrix`

package, or`simple_triplet_matrix`

from the`slam`

package. **obj**- The linear objective vector (the
`c`

vector in the problem statement above). You must specify one value for each column of`A`

. **sense**- The senses of the linear constraints. Allowed values are
`'='`

,`'<='`

, or`'>='`

. You must specify one value for each row of`A`

. **rhs**- The right-hand side vector for the linear constraints (the
vector in the problem statement above). You must specify
one value for each row of
`A`

. **lb (optional)**- The lower bound vector. When present, you must
specify one value for each column of
`A`

. When absent, each variable has a lower bound of 0. **ub (optional)**- The upper bound vector. When present, you must
specify one value for each column of
`A`

. When absent, the variables have infinite upper bounds. **vtype (optional)**- The variable type vector. This vector is used
to capture variable integrality constraints. Allowed values are
`'C'`

(continuous),`'B'`

(binary),`'I'`

(integer),`'S'`

(semi-continuous), or`'N'`

(semi-integer). Binary variables must be either 0 or 1. Integer variables can take any integer value between the specified lower and upper bounds. Semi-continuous variables can take any value between the specified lower and upper bounds, or a value of zero. Semi-integer variables can take any integer value between the specified lower and upper bounds, or a value of zero. When present, you must specify one value for each column of`A`

. When absent, each variable is treated as being continuous. **modelsense (optional)**- The optimization sense. Allowed values
are 'min' (minimize) or 'max' (maximize). When absent, the default
optimization sense is minimization.
**modelname (optional)**- The name of the model. The name appears
in the Gurobi log, and when writing a model to a file.
**objcon (optional)**- The constant offset in the objective function
(
in the problem statement above).
**vbasis (optional)**- The variable basis status vector. Used to
provide an advanced starting point for the simplex algorithm. You
would generally never concern yourself with the contents of this
array, but would instead simply pass it from the result of a
previous optimization run to the input of a subsequent run. When
present, you must specify one value for each column of
`A`

. **cbasis (optional)**- The constraint basis status vector. Used to
provide an advanced starting point for the simplex algorithm.
Consult the
`vbasis`

description for details. When present, you must specify one value for each row of`A`

. **Q (optional)**- The quadratic objective matrix. When present,
`Q`

must be a square matrix whose row and column counts are equal to the number of columns in`A`

. The Q matrix can be dense or sparse. Sparse matrices should be built using either`sparseMatrix`

from the`Matrix`

package, or`simple_triplet_matrix`

from the`slam`

package. **cones (optional)**- Second-order cone constraints. A list of
lists. Each member list defines a single cone constraint:
. The first integer in the list gives the column
index for variable , and the remainder give the column
indices for the variables.
**quadcon (optional)**- The quadratic constraints. A list of lists.
When present, each entry in the list defines a single quadratic
constraint:
. The
`Qc`

matrix must be a square matrix whose row and column counts are equal to the number of columns of`A`

. The matrix associated with quadratic constraint should be stored in`model$quadcon[[i]]$Qc`

. The optional`q`

vector defines the linear terms in the constraint. If present, you must specify one value for`q`

for each column of`A`

. It is stored in`model$quadcon[[i]]$q`

. The scalar`beta`

defines the right-hand side of the constraint. It is stored in`model$quadcon[[i]]$rhs`

. **sos (optional)**- The Special Ordered Set (SOS) constraints. A
list of lists. When present, each entry in the list defines a
single SOS constraint. A SOS constraint can be of type 1 or 2. The
type of SOS constraint is specified
via
`model$sos[[i]]$type`

. A type 1 SOS constraint is a set of variables for which at most one variable in the set may take a value other than zero. A type 2 SOS constraint is an ordered set of variables where at most two variables in the set may take non-zero values. If two take non-zeros values, they must be contiguous in the ordered set. The members of an SOS constraint are specified by placing their indices in vector`model$sos[[i]]$index`

. Weights associated with SOS members are provided in vector`model$sos[[i]]$weight`

. **pwlobj (optional)**- The piecewise-linear objective functions. A
list of lists. When present, each entry in the list defines a
piecewise-linear objective function of a single variable. The
index of the variable whose objective function is being defined
is stored in
`model$pwlobj[[i]]$var`

. The values for the points that define the piecewise-linear function are stored in

`model$pwlobj[[i]]$x`

. The values in the vector must be in non-decreasing order. The values for the points that define the piecewise-linear function are stored in`model$pwlobj[[i]]$y`

. **start (optional)**- The MIP start vector. The MIP solver will
attempt to build an initial solution from this vector. When
present, you must specify a start value for each variable. Note
that you can set the start value for a variable to
`NA`

, which instructs the MIP solver to try to fill in a value for that variable.

`gurobi()`

function will return an error.
Below is an example that demonstrates the construction of a simple optimization model:

model <- list() model$A <- matrix(c(1,1,0,0,1,1), nrow=2, byrow=T) model$obj <- c(1,1,2) model$modelsense <- "max" model$rhs <- c(1,1) model$sense <- c('<=', '<=')

You can also build `A`

as a sparse matrix,
using either `sparseMatrix`

or `simple_triplet_matrix`

:

model$A <- spMatrix(2, 3, c(1, 1, 2, 2), c(1, 2, 2, 3), c(1, 1, 1, 1)) model$A <- simple_triplet_matrix(c(1, 1, 2, 2), c(1, 2, 2, 3), c(1, 1, 1, 1))

Note that the Gurobi interface allows you to specify a scalar value
for any of the array-valued components. The specified value will be
expanded to an array of the appropriate size, with each component of
the array equal to the scalar (e.g., `model$rhs <- 1`

would be
equivalent to `model$rhs <- c(1,1)`

in the example).

The parameter list

The optional `params`

argument to the `gurobi()`

function
is also a list of named components. For each component, the name
should be the name of a Gurobi parameter, and the associated value
should be the desired value of that parameter. Gurobi parameters
allow users to modify the default behavior of the Gurobi optimization
algorithms. You can find a complete list of the available Gurobi
parameters here.

To create a list that would set the Gurobi
Method parameter to 2 and the
ResultFile parameter
parameter to `model.mps`

,
you would do the following:

params <- list(Method=2, ResultFile='model.mps')

We should say a bit more about the
`ResultFile`

parameter. If this parameter is set, the optimization
model that is eventually passed to Gurobi will also be output to the
specified file. The filename suffix should be one of `.mps`

,
`.lp`

, `.rew`

, or `.rlp`

,
to indicate the desired file format (see the
file formats
section for details on Gurobi file formats).

The optimization result

The `gurobi()`

function returns a list, with the various results
of the optimization stored in its named components. The specific
results that are available depend on the type of model that was
solved, and the status of the optimization.
The following is a list of components that might be available
in the result list. We'll discuss the circumstances under which
each will be available after presenting the list.

**status**- The status of the optimization, returned as a string.
The desired result is
`"OPTIMAL"`

, which indicates that an optimal solution to the model was found. Other status are possible, for example if the model has no feasible solution or if you set a Gurobi parameter that leads to early solver termination. See the Status Code section for further information on the Gurobi status codes. **objval**- The objective value of the computed solution.
**runtime**- The elapsed wall-clock time (in seconds) for the
optimization.
**x**- The computed solution. This array contains one entry for
each column of
`A`

. **slack**- Constraint slacks for the computed solution. This array
contains one entry for each row of
`A`

. **pi**- Dual values for the computed solution (also known as
*shadow prices*). This array contains one entry for each row of`A`

. **rc**- Variable reduced costs for the computed solution. This
array contains one entry for each column of
`A`

. **vbasis**- Variable basis status values for the computed optimal
basis. You generally should not concern yourself with the contents
of this array. If you wish to use an advanced start later, you
would simply copy the
`vbasis`

and`cbasis`

arrays into the corresponding components for the next model. This array contains one entry for each column of`A`

. **cbasis**- Constraint basis status values for the computed optimal
basis. This array contains one entry for each row of
`A`

. **objbound**- Best available bound on solution (lower bound for
minimization, upper bound for maximization).
**itercount**- Number of simplex iterations performed.
**baritercount**- Number of barrier iterations performed.
**nodecount**- Number of branch-and-cut nodes explored.

`status`

component will be present in all cases. It
indicates whether Gurobi was able to find a proven optimal solution to
the model. In cases where a solution to the model was found, optimal
or otherwise, the `objval`

, `x`

, and `slack`

components will be present. For linear and quadratic programs, if a
solution is available, then the `pi`

and `rc`

components
will also be present. Finally, if the final solution is a
*basic*solution (computed by simplex), then

`vbasis`

and
`cbasis`

will be present.
The following is an example of how the results of the `gurobi()`

call might be extracted and output:

result <- gurobi(model, params) print(result$objval) print(result$x)