CGAL 5.0.2 - Linear and Quadratic Programming Solver
NonnegativeLinearProgram Concept Reference

## Definition

A model of NonnegativeLinearProgram 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, \\ & & \qpx \geq 0 \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 $$\{\leq, =, \geq\}$$,
• $$\qpc$$ 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::Nonnegative_linear_program_from_iterators<A_it, B_it, R_it, FL_it, L_it, FU_it, U_it, D_it, C_it>

CGAL::Quadratic_program_from_mps<NT>

CGAL::Nonnegative_linear_program_from_iterators<A_it, B_it, R_it, C_it>

The value types of all iterator types (nested iterator types, respectively, for A_iterator) must be convertible to some common IntegralDomain ET.

Has Models:
CGAL::Quadratic_program<NT>
QuadraticProgram
LinearProgram
NonnegativeQuadraticProgram

## 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 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...

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.

## ◆ 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.

## ◆ R_iterator

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

The value type of R_iterator is CGAL::Comparison_result.

## ◆ get_a()

 A_iterator NonnegativeLinearProgram::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) a random access iterator for column $$j$$.

## ◆ get_b()

 B_iterator NonnegativeLinearProgram::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 NonnegativeLinearProgram::get_c ( ) const

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

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

## ◆ get_r()

 R_iterator NonnegativeLinearProgram::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 $$\geq$$.