LinearProgram Concept Reference

Definition

A model of LinearProgram describes a linear program of the form.

$\newcommand{\qprel}{\gtreqless} \newcommand{\qpx}{\mathbf{x}} \newcommand{\qpl}{\mathbf{l}} \newcommand{\qpu}{\mathbf{u}} \newcommand{\qpc}{\mathbf{c}} \newcommand{\qpb}{\mathbf{b}} \newcommand{\qpy}{\mathbf{y}} \newcommand{\qpw}{\mathbf{w}} \newcommand{\qplambda}{\mathbf{\lambda}}$

$$\begin{eqnarray*} \mbox{(QP)}& \mbox{minimize} & \qpc^{T}\qpx+c_0 \\ &\mbox{subject to} & A\qpx\qprel \qpb, \\ & & \qpl \leq \qpx \leq \qpu \end{eqnarray*}$$

in $n$ real variables $\qpx=(x_0,\ldots,x_{n-1})$. Here,

• $A$ is an $m\times n$ matrix (the constraint matrix),
• $\qpb$ is an $m$-dimensional vector (the right-hand side),

• $\qprel$ is an $m$-dimensional vector of relations from $\{\leq, =, \geq\}$,

• $\qpl$ is an $n$-dimensional vector of lower bounds for $\qpx$, where $l_j\in\mathbb{R}\cup\{-\infty\}$ for all $j$

• $\qpu$ is an $n$-dimensional vector of upper bounds for $\qpx$, where $u_j\in\mathbb{R}\cup\{\infty\}$ for all $j$

• $\qpc$ is an $n$-dimensional vector (the linear objective function), and

• $c_0$ is a constant.

The description is given by appropriate random-access iterators over the program data, see below. The program therefore comes in dense representation which includes zero entries.

Has Models:

CGAL::Quadratic_program<NT>

CGAL::Quadratic_program_from_mps<NT>

CGAL::Linear_program_from_iterators<A_it, B_it, R_it, FL_it, L_it, FU_it, U_it, D_it, C_it>

and the other concepts

See also

QuadraticProgram

NonnegativeQuadraticProgram

NonnegativeLinearProgram

Types
typedef unspecified_type	A_iterator
	A random access iterator type to go columnwise over the constraint matrix $A$. More...
typedef unspecified_type	B_iterator
	A random access iterator type to go over the entries of the right-hand side $\qpb$.
typedef unspecified_type	R_iterator
	A random access iterator type to go over the relations $\qprel$. More...
typedef unspecified_type	FL_iterator
	A random access iterator type to go over the existence (finiteness) of the lower bounds $l_j, j=0,\ldots,n-1$. More...
typedef unspecified_type	L_iterator
	A random acess iterator type to go over the entries of the lower bound vector $\qpl$.
typedef unspecified_type	UL_iterator
	A random access iterator type to go over the existence (finiteness) of the upper bounds $u_j, j=0,\ldots,n-1$. More...
typedef unspecified_type	U_iterator
	A random acess iterator type to go over the entries of the upper bound vector $\qpu$.
typedef unspecified_type	C_iterator
	A random access iterator type to go over the entries of the linear objective function vector $c$.

Operations
int	get_n () const
	returns the number $n$ of variables (number of columns of $A$) in lp.
int	get_m () const
	returns the number $m$ of constraints (number of rows of $A$) in lp.
A_iterator	get_a () const
	returns an iterator over the columns of $A$. More...
B_iterator	get_b () const
	returns an iterator over the entries of $\qpb$. More...
R_iterator	get_r () const
	returns an iterator over the entries of $\qprel$. More...
FL_iterator	get_fl () const
	returns an iterator over the existence of the lower bounds $l_j, j=0,\ldots,n-1$. More...
L_iterator	get_l () const
	returns an iterator over the entries of $\qpl$. More...
FU_iterator	get_fu () const
	returns an iterator over the existence of the upper bounds $u_j, j=0,\ldots,n-1$. More...
L_iterator	get_u () const
	returns an iterator over the entries of $\qpu$. More...
C_iterator	get_c () const
	returns an iterator over the entries of $\qpc$. More...
std::iterator_traits< C_iterator >::value_type	get_c0 () const
	returns the constant term $c_0$ of the objective function.

Member Typedef Documentation

A_iterator

typedef unspecified_type LinearProgram::A_iterator

A random access iterator type to go columnwise over the constraint matrix $A$.

The value type is a random access iterator type for an individual column that goes over the entries in that column.

FL_iterator

typedef unspecified_type LinearProgram::FL_iterator

A random access iterator type to go over the existence (finiteness) of the lower bounds $l_j, j=0,\ldots,n-1$.

The value type of FL_iterator is bool.

R_iterator

typedef unspecified_type LinearProgram::R_iterator

A random access iterator type to go over the relations $\qprel$.

The value type of R_iterator is CGAL::Comparison_result.

UL_iterator

typedef unspecified_type LinearProgram::UL_iterator

A random access iterator type to go over the existence (finiteness) of the upper bounds $u_j, j=0,\ldots,n-1$.

The value type of UL_iterator is bool.

Member Function Documentation

get_a()

A_iterator LinearProgram::get_a

(

)

const

returns an iterator over the columns of $A$.

The corresponding past-the-end iterator is get_a()+get_n(). For $j=0,\ldots,n-1$, *(get_a()+j) is a random access iterator for column $j$.

get_b()

B_iterator LinearProgram::get_b

(

)

const

returns an iterator over the entries of $\qpb$.

The corresponding past-the-end iterator is get_b()+get_m().

get_c()

C_iterator LinearProgram::get_c

(

)

const

returns an iterator over the entries of $\qpc$.

The corresponding past-the-end iterator is get_c()+get_n().

get_fl()

FL_iterator LinearProgram::get_fl

(

)

const

returns an iterator over the existence of the lower bounds $l_j, j=0,\ldots,n-1$.

The corresponding past-the-end iterator is get_fl()+get_n(). If *(get_fl()+j) has value $true$, the variable $x_j$ has a lower bound given by *(get_l()+j), otherwise it has no lower bound.

get_fu()

FU_iterator LinearProgram::get_fu

(

)

const

returns an iterator over the existence of the upper bounds $u_j, j=0,\ldots,n-1$.

The corresponding past-the-end iterator is get_fu()+get_n(). If *(get_fu()+j) has value $true$, the variable $x_j$ has an upper bound given by *(get_u()+j), otherwise it has no upper bound.

get_l()

L_iterator LinearProgram::get_l

(

)

const

returns an iterator over the entries of $\qpl$.

The corresponding past-the-end iterator is get_l()+get_n(). If *(get_fl()+j) has value $false$, the value *(get_l()+j) is not accessed.

Precondition

if both *(get_fl()+j) and *(get_fu()+j) have value $true$, then *(get_l()+j) <= *(get_u()+j).

get_r()

R_iterator LinearProgram::get_r

(

)

const

returns an iterator over the entries of $\qprel$.

The corresponding past-the-end iterator is get_r()+get_m(). The value CGAL::SMALLER stands for $\leq$, CGAL::EQUAL stands for $=$, and CGAL::LARGER stands for $\geq$.

get_u()

L_iterator LinearProgram::get_u

(

)

const

returns an iterator over the entries of $\qpu$.

The corresponding past-the-end iterator is get_u()+get_n(). If *(get_fu()+j) has value $false$, the value *(get_u()+j) is not accessed.

Precondition

if both *(get_fl()+j) and *(get_fu()+j) have value $true$, then *(get_l()+j) <= *(get_u()+j).