QuadraticProgram Concept Reference

Definition

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

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

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>

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 $A$.
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 $\qprel$.
typedef unspecified_type	FL_iterator
	A random access iterator type to go over the existence (finiteness) of the lower bounds $l_j, j=0,\ldots,n-1$.
typedef unspecified_type	L_iterator
	A random access 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 $u_j, j=0,\ldots,n-1$.
typedef unspecified_type	U_iterator
	A random access 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 $2D$.
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 $A$.
B_iterator	get_b () const
	returns an iterator over the entries of $\qpb$.
R_iterator	get_r () const
	returns an iterator over the entries of $\qprel$.
FL_iterator	get_fl () const
	returns an iterator over the existence of the lower bounds $l_j, j=0,\ldots,n-1$.
L_iterator	get_l () const
	returns an iterator over the entries of $\qpl$.
FU_iterator	get_fu () const
	returns an iterator over the existence of the upper bounds $u_j, j=0,\ldots,n-1$.
L_iterator	get_u () const
	returns an iterator over the entries of $\qpu$.
D_iterator	get_d () const
	returns an iterator over the rows of $2D$.
C_iterator	get_c () const
	returns an iterator over the entries of $\qpc$.
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 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.

D_iterator

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.

FL_iterator

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.

R_iterator

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.

UL_iterator

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

get_a()

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 $j$.

get_b()

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().

get_c()

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().

get_d()

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 $D$.

get_fl()

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.

get_fu()

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.

get_l()

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).

get_r()

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 $\geq$.

get_u()

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).