# Operations on LP objects

## `AddConstraints(L, lhs, rhs): LP, Mtrx, Mtrx`

```magma
Rel: MonStgElt                    Default: "eq"
```

Add some constraints to the LP problem $L$. All constraints will have the same relation, given by `Rel`, which may be set to `"eq"` for strict equality (the default), `"le"` for less-or-equal constraints, or `"ge"` for greater-or-equal constraints.

Constraints are of the form

$$
\sum_{j = 1}^{n}\hbox{\tt lhs}_{ij}\quad \hbox{\tt Rel}\quad \hbox{\tt rhs}_{i1}
$$

where $lhs$ and $rhs$ are described in Section [Explicit LP Solving Functions](instant-lp.md#lp-explicit).

## `NumberOfConstraints(L): LP -> RngIntElt`

The number of constraints in the LP problem $L$.

## `NumberOfVariables(L): LP -> RngIntElt`

The number of variables in the LP problem $L$.

## `EvaluateAt(L, p): LP, Mtrx -> RngIntElt`

Evaluate the objective function of the LP problem $L$ at the point $p$ given by a matrix.

## `Constraint(L, n): LP, RngIntElt -> Mtrx, Mtrx, RngIntElt`

The LHS, RHS and relation ($-1$ for $\le$, $0$ for $=$, $1$ for $\ge$) of the $n$-th constraint of the LP problem $L$.

## `IntegerSolutionVariables(L): LP -> SeqEnum`

Sequence of indices of the variables in the LP problem $L$ to be solved in integers.

## `ObjectiveFunction(L): LP -> Mtrx`

The objective function of the LP problem $L$.

## `IsMaximisingFunction(L): LP -> BoolElt`

Returns `true` if the LP problem $L$ is set to maximise its objective function, `false` if set to minimise.

## `RemoveConstraint(L, n): LP, RngIntElt`

Remove the $n$-th constraint from the LP problem $L$.

## `SetIntegerSolutionVariables(L, I, m): LP, SeqEnum[RngIntElt], BoolElt`

Set the variables of the LP problem $L$ indexed by elements of the sequence $I$ to be solved in integers if $m$ is `true`, or in the usual ring if `false`.

## `SetLowerBound(L, n, b): LP, RngIntElt, RngElt`

Set the lower bound on the $n$-th variable in the LP problem $L$ to $b$.

Note that for all LP problems in Magma there is an implicit constraint that all variables are $\geq 0$. This constraint is overridden if a lower bound is specified by using this function (e.g., specifying a lower bound of $-5$ works as expected), but the lower bound can currently not be completely removed.

## `SetMaximiseFunction(L, m): LP, BoolElt`

Set the LP problem $L$ to maximise its objective function if $m$ is `true`, or to minimise the objective function if $m$ is false.

## `SetObjectiveFunction(L, F): LP, Mtrx`

Set the objective function of the LP problem $L$ to the matrix $F$.

## `SetUpperBound(L, n, b): LP, RngIntElt, RngElt`

Set the upper bound on the $n$-th variable in the LP problem $L$ to $b$.

## `Solution(L): LP -> Mtrx, RngIntElt`

Solve the LP problem $L$; returns a point representing an optimal solution, and an integer representing the state of the solution.

## `UnsetBounds(L): LP`

Remove any bounds on all variables in the LP problem $L$.

Note that this reactivates the implicit constraint that all variables are $\geq 0$.

## `Example: Filling LP Object (ex-b5bf57)`

We use an LP object to solve the LP maximising $F(x,y) = 3x + 13y$ subject to constraints $2x + 9y <= 40$ $11x -8y <= 82$

```magma
> R := RealField( );
> L := LPProcess(R, 2);
> SetObjectiveFunction(L, Matrix(R, 1, 2, [3,13]));
> lhs := Matrix(R, 2, 2, [2, 9, 11, -8]);
> rhs := Matrix(R, 2, 1, [40, 82]);
> AddConstraints(L, lhs, rhs : Rel := "le");
> SetMaximiseFunction(L, true);
> L;
LP <Real Field, 2 variables>
Maximising objective function: [ 3 13]
Subject to constraints:
1 : [2 9] <= [40]
2 : [11 -8] <= [82]
Variables bounded above by: [ ]
Variables bounded below by: [ ]
Solving in integers for variables [ ]
> Solution(L);
[9.199999999999999289 2.400000000000000355]
0

```

Now, we place some bounds on y:

```magma
> SetUpperBound(L, 2, R!2);
> SetLowerBound(L, 2, R!1);
> Solution(L);
[8.909090909090908283 2.000000000000000000]
0

```

And find integer solutions:

```magma
> SetIntegerSolutionVariables(L, [1,2], true);
> Solution(L);
[8.000000000000000000 2.000000000000000000]
0

```

Now, removing the 2nd constraint:

```magma
> RemoveConstraint(L, 2);
> L;
LP <Real Field, 2 variables>
Maximising objective function: [ 3 13]
Subject to constraints:
1 : [2 9] <= [40]
Variables bounded above by: [ 2:2 ]
Variables bounded below by: [ 2:1 ]
Solving in integers for variables [ 1, 2 ]
> Solution(L);
[11.00000000000000000 2.000000000000000000]
0

```

And removing the restriction to Integer values for y,

```magma
> SetIntegerSolutionVariables(L, [2], false);
> Solution(L);
[15.00000000000000000 1.111111111111111160]
0

```