# The model argument

### The model argument

Model variables store optimization problems (as described in the`problem`statement).

Models can be built in a number of ways. You can
populate the appropriate named components of the
`model` list using standard R routines.
You can also read a model from a file, using
`gurobi_read`.
A few API functions (
`gurobi_feasrelax` and
`gurobi_relax`) also return models.

Note that all matrix named components within the
`model` variable 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.

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:

Commonly used named components:

**A**- The linear constraint matrix.
**obj (optional)**- The linear objective vector (the
`c`vector in the`problem`statement). When present, you must specify one value for each column of`A`. When absent, each variable has a default objective coefficient of 0. **sense (optional)**- The senses of the linear constraints. Allowed
values are
`'='`,`'<'`, or`'>'`. You must specify one value for each row of`A`, or a single value to specify that all constraints have the same sense. When absent, all senses default to`'<'`. **rhs (optional)**- The right-hand side vector for the linear
constraints ( in the
`problem`statement). You must specify one value for each row of`A`. When absent, the right-hand side vector defaults to the zero vector. **lb (optional)**- The lower bound vector. When present, you must
specify one value for each column of
`A`. When absent, each variable has a default 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 types. 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`, or a single value to specify that all variables have the same type. When absent, each variable is treated as being continuous. Refer to this section for more information on variable types. **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). **varnames (optional)**- The variable names vector. A character vector.
When present, each element of this vector defines the name of a
variable. You must specify a name for each column of
`A`. **constrnames (optional)**- The constraint names vector. A
character vector. When present, each element of the vector defines the
name of a constraint. You must specify a name for each row of
`A`.

Quadratic objective and constraint named components:

**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`. **quadcon (optional)**- The quadratic constraints. A list of lists. When
present, each element in
`quadcon`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`. It is stored in`model$quadcon[[i]]$Qc`.The

`q`vector defines the linear terms in the constraint. It must specify a value for each column of`A`. It is stored in`model$quadcon[[i]]$q`.The scalar

`beta`is stored in`model$quadcon[[i]]$rhs`. It defines the right-hand side value for the constraint.The optional

`sense`string defines the sense of the quadratic constrint. Allowed values are`'<'`,`'='`or`'>'`. If not present, the default sense is`'<'`. It is stored in`model$quadcon[[i]]$sense`.The optional

`name`string defines the name of the quadratic constraint. It is stored in`model$quadcon[[i]]$name`.

SOS constraint named components:

**sos (optional)**- The Special Ordered Set (SOS) constraints.
A list of lists. When present, each entry in
`sos`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 where 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`. Please refer to this section for details on SOS constraints.

Multi-objective named components:

**multiobj (optional)**- Multi-objective specification for the model.
A list of lists. When present, each entry in
`multiobj`defines a single objective of a multi-objective problem. Please refer to the Multiple Objectives section for more details on multi-objective optimization. Each objective may have the following named components:**objn**- Specified via
`model$multiobj[[i]]$objn`. This is the*i*-th objective vector. **objcon (optional)**- Specified via
`model$multiobj[[i]]$objcon`. If provided, this is the*i*-th objective constant. The default value is 0. **priority (optional)**- Specified via
`model$multiobj[[i]]$priority`. If provided, this value is the*hierarchical*priority for this objective. The default value is 0. **weight (optional)**- Specified via
`model$multiobj[[i]]$weight`. If provided, this value is the multiplier used when aggregating objectives. The default value is 1.0. **reltol (optional)**- Specified via
`model$multiobj[[i]]$reltol`. If provided, this value specifies the relative objective degradation when doing hierarchical multi-objective optimization. The default value is 0. **abstol (optional)**- Specified via
`model$multiobj[[i]]$abstol`. If provided, this value specifies the absolute objective degradation when doing hierarchical multi-objective optimization. The default value is 0. **name (optional)**- Specified via
`model$multiobj[[i]]$name`. If provided, this string specifies the name of the*i*-th objective function.

Note that when multiple objectives are present, the

`result$objval`named component that is returned in the result of an optimization call will be a vector of the same length as`model$multiobj`.A multi-objective model can't have other objectives. Thus, combining

`model$multiobj`with any of`model$obj`,`model$objcon`,`model$pwlobj`, or`model$Q`is an error.

General constraint named components:

The list of lists described below are used to add *general constraints* to a model.

Mathematical programming has traditionally defined a set of fundamental constraint types: variable bound constraints, linear constraints, quadratic constraints, integrality constraints, and SOS constraints. These are typically treated directly by the underlying solver (although not always), and are fundamental to the overall algorithm.

Gurobi accepts a number of additional constraint types, which we
collectively refer to as *general constraints*. These are
typically *not* treated directly by the solver. Rather, they
are transformed by presolve into mathematically equivalent sets of
constraints (and variables), chosen from among the fundamental types
listed above. These general constraints are provided as a convenience
to users. If such constraints appear in your model, but if you prefer
to reformulate them yourself using fundamental constraint types
instead, you can certainly do so. However, note that Gurobi can
sometimes exploit information contained in the other constraints in
the model to build a more efficient formulation than what you might
create.

The additional constraint types that fall under this *general constraint* umbrella are:

- MAX (genconmax): set a decision variable equal to the maximum value from among a set of decision variables
- MIN (genconmin): set a decision variable equal to the minimum value from among a set of decision variables
- ABS (genconabs): set a decision variable equal to the absolute value of some other decision variable
- AND (genconand): set a binary variable equal to one if and only if all of a set of binary decision variables are equal to one
- OR (genconor): set a binary variable equal to one if and only if at least one variable out of a set of binary decision variables is equal to one
- INDICATOR (genconind): whenever a given binary variable takes a certain value, then the given linear constraint must be satisfied

Please refer to this section for additional details on general constraints.

**genconmax (optional)**- A list of lists. When present, each entry in
`genconmax`defines a MAX general constraint of the form

Each entry may have the following named components:**resvar**- Specified via
`model$genconmax[[i]]$resvar`. Index of the variable in the left-hand side of the constraint. **vars**- Specified via
`model$genconmax[[i]]$vars`, it is a vector of indices of variables in the right-hand side of the constraint. **con (optional)**- Specified via
`model$genconmax[[i]]$con`. When present, specifies the constant on the left-hand side. Default value is . **name (optional)**- Specified via
`model$genconmax[[i]]$name`. When present, specifies the name of the -th MAX general constraint.

**genconmin (optional)**- A list of lists. When present, each entry in
`genconmax`defines a MIN general constraint of the form

Each entry may have the following named components:**resvar**- Specified via
`model$genconmin[[i]]$resvar`. Index of the variable in the left-hand side of the constraint. **vars**- Specified via
`model$genconmin[[i]]$vars`, it is a vector of indices of variables in the right-hand side of the constraint. **con (optional)**- Specified via
`model$genconmin[[i]]$con`. When present, specifies the constant on the left-hand side. Default value is . **name (optional)**- Specified via
`model$genconmin[[i]]$name`. When present, specifies the name of the -th MIN general constraint.

**genconabs (optional)**- A list of lists. When present, each entry in
`genconmax`defines an ABS general constraint of the form

Each entry may have the following named components:**resvar**- Specified via
`model$genconabs[[i]]$resvar`. Index of the variable in the left-hand side of the constraint. **argvar**- Specified via
`model$genconabs[[i]]$argvar`. Index of the variable in the right-hand side of the constraint. **name (optional)**- Specified via
`model$genconabs[[i]]$name`. When present, specifies the name of the -th ABS general constraint.

**genconand (optional)**- A list of lists. When present, each entry in
`genconand`defines an AND general constraint of the form

Each entry may have the following named components:**resvar**- Specified via
`model$genconand[[i]]$resvar`. Index of the variable in the left-hand side of the constraint. **vars**- Specified via
`model$genconand[[i]]$vars`, it is a vector of indices of variables in the right-hand side of the constraint. **name (optional)**- Specified via
`model$genconand[[i]]$name`. When present, specifies the name of the -th AND general constraint.

**genconor (optional)**- A list of lists. When present, each entry in
`genconor`defines an OR general constraint of the form

Each entry may have the following named components:**resvar**- Specified via
`model$genconor[[i]]$resvar`. Index of the variable in the left-hand side of the constraint. **vars**- Specified via
`model$genconor[[i]]$vars`, it is a vector of indices of variables in the right-hand side of the constraint. **name (optional)**- Specified via
`model$genconor[[i]]$name`. When present, specifies the name of the -th OR general constraint.

**genconind (optional)**- A list of lists. When present, each entry in
`genconind`defines an INDICATOR general constraint of the form

This constraint states that when the binary variable takes the value`binval`then the linear constraint must hold. Note that`sense`is one of`'='`,`'<'`, or`'>'`for equality (), less than or equal () or greater than or equal () constraints. Each entry may have the following named components:**binvar**- Specified via
`model$genconind[[i]]$binvar`. Index of the implicating binary variable. **binval**- Specified via
`model$genconind[[i]]$binval`. Value for the binary variable that forces the following linear constraint to be satisfied. It can be either 0 or 1. **a**- Specified via
`model$genconind[[i]]$a`. Vector of coefficients of variables participating in the implied linear constraint. You must specify a value for`a`for each column of`A`. **sense**- Specified via
`model$genconind[[i]]$sense`. Sense of the implied linear constraint. Must be one of`'='`,`'<'`, or`'>'`. **rhs**- Specified via
`model$genconind[[i]]$rhs`. Right-hand side value of the implied linear constraint. **name (optional)**- Specified via
`model$genconind[[i]]$name`. When present, specifies the name of the -th INDICATOR general constraint.

Advanced named components:

**pwlobj (optional)**- The piecewise-linear objective functions.
A list of lists. When present, each entry in
`pwlobj`defines a piecewise-linear objective function for 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`. **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
vector, 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`. **varhintval (optional)**- A set of user hints. If you know that a
variable is likely to take a particular value in high quality
solutions of a MIP model, you can provide that value as a hint. You
can also (optionally) provide information about your level of
confidence in a hint with the
`varhintpri`named component. If present, you must specify one value for each column of`A`. Use a value of`NA`for variables where no such hint is known. For more details, please refer to the VarHitVal attribute documentation. **varhintpri (optional)**- Priorities on user hints. After
providing variable hints through the
`varhintval`list, you can optionally also provide hint priorities to give an indication of your level of confidence in your hints. If present, you must specify a value for each column of`A`. For more details, please refer to the VarHintPri attribute documentation. **branchpriority (optional)**- Variable branching priority. If
present, the value of this attribute is used as the primary criteria
for selecting a fractional variable for branching during the MIP
search. Variables with larger values always take priority over those
with smaller values. Ties are broken using the standard branch
variable selection criteria.
If present, you must specify one value
for each column of
`A`. **pstart (optional)**- The current simplex start vector. If you set
`pstart`values for every variable in the model and`dstart`values for every constraint, then simplex will use those values to compute a warm start basis. For more details, please refer to the PStart attribute documentation. **dstart (optional)**- The current simplex start vector. If you set
`dstart`values for every linear constraint in the model and`pstart`values for every variable, then simplex will use those values to compute a warm start basis. For more details, please refer to the DStart attribute documentation. **lazy (optional)**- Determines whether a linear constraint is
treated as a
*lazy constraint*. If present, you must specify one value for each row of`A`. For more details, please refer to the Lazy attribute documentation. **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. **partition (optional)**- The MIP variable partition number, which
is used by the MIP solution improvement heuristic. If present,
you must specify one value for each variable of
`A`. For more details, please refer to the Partition attribute documentation.

If any of the mandatory components listed above are missing, the
`gurobi()` function will return an error.

Below is an example that demonstrates the construction of a simple
optimization model:

`model <- list()
model$A <- matrix(c(1,2,3,1,1,0), nrow=2, byrow=T)
model$obj <- c(1,1,1)
model$modelsense <- 'max'
model$rhs <- c(4,1)
model$sense <- c('<', '>')
`

You can also build `A` as a sparse matrix, using
either `sparseMatrix` or `simple_triplet_matrix`:

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

Note that the Gurobi R interface allows you to specify a scalar value
for most 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$obj <- 1` would
be equivalent to `model$obj <- c(1,1,1)` in the example).