QuadraticProgram

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QuadraticProgram

Definition

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

(QP)	minimize	x^TDx+c^Tx+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 ∈ ℝ ∪ {-∞} for all j
u is an n-dimensional vector of upper bounds for x, where u_j ∈ ℝ ∪ {∞} 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

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>

Types

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.

QuadraticProgram::B_iterator
	A random access iterator type to go over the entries of the right-hand side b.

QuadraticProgram::R_iterator
	A random access iterator type to go over the relations ⋛ . The value type of R_iterator is CGAL::Comparison_result.

QuadraticProgram::FL_iterator
	A random access iterator type to go over the existence (finiteness) of the lower bounds l_j, j=0, … ,n-1. The value type of FL_iterator is bool.

QuadraticProgram::L_iterator
	A random acess iterator type to go over the entries of the lower bound vector l.

QuadraticProgram::UL_iterator
	A random access iterator type to go over the existence (finiteness) of the upper bounds u_j, j=0, … ,n-1. The value type of UL_iterator is bool.

QuadraticProgram::U_iterator
	A random acess iterator type to go over the entries of the upper bound vector u.

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.

QuadraticProgram::C_iterator
	A random access iterator type to go over the entries of the linear objective function vector c.

Operations

int

qp.get_n () const

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

int

qp.get_m () const

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

A_iterator

qp.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, … ,n-1, *(get_a()+j) is a random access iterator for column j.

B_iterator

qp.get_b () const

returns an iterator over the entries of b. The corresponding past-the-end iterator is get_b()+get_m().

R_iterator

qp.get_r () const

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 () const

returns an iterator over the existence of the lower bounds l_j, j=0, … ,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.

L_iterator

qp.get_l () const

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 () const

returns an iterator over the existence of the upper bounds u_j, j=0, … ,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

qp.get_u () const

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 () const

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 () const

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 () const

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.