Definition

A model of QuadraticProgram describes a convex quadratic program of the form.

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

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

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 $j$
$D$ is a symmetric positive-semidefinite $n\times n$ matrix (the quadratic objective function),
$\qpc$ 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::Quadratic_program_from_iterators<A_it, B_it, R_it, FL_it, L_it, FU_it, U_it, D_it, C_it>

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

See Also: NonnegativeQuadraticProgram; LinearProgram; 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	D_iterator
	A random access iterator type to go rowwise over the matrix $2D$ . 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 `qp`.

int	get_m () const
	returns the number $m$ of constraints (number of rows of $A$ ) in `qp`.

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...

D_iterator	get_d () const
	returns an iterator over the rows of $2D$ . 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

typedef unspecified_type QuadraticProgram::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.

typedef unspecified_type QuadraticProgram::D_iterator

A random access iterator type to go rowwise over the matrix $2D$ .

The value type is a random access iterator type for an individual row that goes over the entries in that row, up to (and including) the entry on the main diagonal.

typedef unspecified_type QuadraticProgram::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.

typedef unspecified_type QuadraticProgram::R_iterator

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

The value type of R_iterator is CGAL::Comparison_result.

typedef unspecified_type QuadraticProgram::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

A_iterator QuadraticProgram::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 $j$ .

B_iterator QuadraticProgram::get_b ( ) const

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

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

C_iterator QuadraticProgram::get_c ( ) const

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

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

D_iterator QuadraticProgram::get_d ( ) const

returns an iterator over the rows of $2D$ .

The corresponding past-the-end iterator is get_d()+get_n(). For $i=0,\ldots,n-1$ , *(get_d()+i) is a random access iterator for the entries in row $i$ below or on the diagonal. The valid range of this iterator is guaranteed to have length $i+1$ but not more. Values to the right of the diagonal are deduced from the symmetry requirement on $D$ .

FL_iterator QuadraticProgram::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.

FU_iterator QuadraticProgram::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.

L_iterator QuadraticProgram::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).

R_iterator QuadraticProgram::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 $\geq$ .

L_iterator QuadraticProgram::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).

Definition

Types

Operations

Member Typedef Documentation

Member Function Documentation