QuadraticProgram Concept Reference

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 \( \qpb\). More...
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 \( \qpl\). More...
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 \( \qpu\). More...
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 \( c\). More...

Operations
int	get_n () const
	returns the number \( n\) of variables (number of columns of \( A\)) in qp. More...
int	get_m () const
	returns the number \( m\) of constraints (number of rows of \( A\)) in qp. More...
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. More...

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::B_iterator

A random access iterator type to go over the entries of the right-hand side \( \qpb\).

typedef unspecified_type QuadraticProgram::C_iterator

A random access iterator type to go over the entries of the linear objective function vector \( c\).

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::L_iterator

A random acess iterator type to go over the entries of the lower bound vector \( \qpl\).

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::U_iterator

A random acess iterator type to go over the entries of the upper bound vector \( \qpu\).

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().

std::iterator_traits<C_iterator>::value_type QuadraticProgram::get_c0

(

)

const

returns the constant term \( c_0\) of the objective function.

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).

int QuadraticProgram::get_m

(

)

const

returns the number \( m\) of constraints (number of rows of \( A\)) in qp.

int QuadraticProgram::get_n

(

)

const

returns the number \( n\) of variables (number of columns of \( A\)) in qp.

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).