QuadraticProgram

Definition

(QP)$$	minimize$$	xTDx+cTx+c0
	subject to$$	Ax ~ b,
		l x u

in $$n real variables $$x=(x₀,...,x_n-1). Here,

• $$A is an $$m × n matrix (the constraint matrix),
• $$b is an $$m-dimensional vector (the right-hand side),
• $$ ~  is an $$m-dimensional vector of relations from $${ , =, },
• $$l is an $$n-dimensional vector of lower bounds for $$x, where $$l_j {- } for all $$j
• $$u is an $$n-dimensional vector of upper bounds for $$x, where $$u_j { } for all $$j
• $$D is a symmetric positive-semidefinite $$n × n matrix (the quadratic objective function),
• $$c is an $$n-dimensional vector (the linear objective function), and
• $$c₀ 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

Types

Operations

int	qp.get_n ()	returns the number $$n of variables (number of columns of $$A) in qp.
int	qp.get_m ()	returns the number $$m of constraints (number of rows of $$A) in qp.
A_iterator	qp.get_a ()	returns an iterator over the columns of $$A. The corresponding past-the-end iterator is get_a()+get_n(). For $$j=0,...,n-1, $$*(get_a()+j) is a random access iterator for column $$j.
B_iterator	qp.get_b ()	returns an iterator over the entries of $$b. The corresponding past-the-end iterator is get_b()+get_m().
R_iterator	qp.get_r ()	returns an iterator over the entries of $$ ~ . The corresponding past-the-end iterator is get_r()+get_m(). The value CGAL::SMALLER stands for $$ , CGAL::EQUAL stands for $$=, and CGAL::LARGER stands for $$ .
FL_iterator	qp.get_fl ()	returns an iterator over the existence of the lower bounds $$lj, j=0,...,n-1. The corresponding past-the-end iterator is get_fl()+get_n(). If *(get_fl()+j) has value $$true, the variable $$xj has a lower bound given by *(get_l()+j), otherwise it has no lower bound.
L_iterator	qp.get_l ()	returns an iterator over the entries of $$l. The corresponding past-the-end iterator is get_l()+get_n(). If *(get_fl()+j) has value $$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)
FU_iterator	qp.get_fu ()	returns an iterator over the existence of the upper bounds $$uj, j=0,...,n-1. The corresponding past-the-end iterator is get_fu()+get_n(). If *(get_fu()+j) has value $$true, the variable $$xj has an upper bound given by *(get_u()+j), otherwise it has no upper bound.
L_iterator	qp.get_u ()	returns an iterator over the entries of $$u. 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 $$*(get_l()+j) *(get_u()+j)
D_iterator	qp.get_d ()	returns an iterator over the rows of $$2D. The corresponding past-the-end iterator is get_d()+get_n(). For $$i=0,...,n-1, $$*(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 $$D.
C_iterator	qp.get_c ()	returns an iterator over the entries of $$c. The corresponding past-the-end iterator is get_c()+get_n().
std::iterator_traits<C_iterator>::value_type
	qp.get_c0 ()	returns the constant term $$c0 of the objective function.

Requirements

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

Quadratic_program<NT>
Quadratic_program_from_mps<NT>
Quadratic_program_from_iterators<A_it, B_it, R_it, FL_it, L_it, FU_it, U_it, D_it, C_it>

and the other conepts

NonnegativeQuadraticProgramInterface
LinearProgramInterface
NonnegativeLinearProgramInterface