mirror of https://github.com/CGAL/cgal
- Free models added;
- figures/program for closest point in intersection of halfspaces
This commit is contained in:
Bernd Gärtner
parent
3a8a8f00d2
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@ -1681,6 +1681,10 @@ Principal_component_analysis/demo/Principal_component_analysis/windows/3d/res/id
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Principal_component_analysis/demo/Principal_component_analysis/windows/3d/res/idb8.bmp -text svneol=unset#image/bmp
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Principal_component_analysis/doc_tex/Principal_component_analysis/fit.eps -text svneol=unset#application/postscript
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Principal_component_analysis/doc_tex/Principal_component_analysis/fit.png -text svneol=unset#image/png
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QP_solver/doc_tex/QP_solver/closest_point.eps -text
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QP_solver/doc_tex/QP_solver/closest_point.fig -text
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QP_solver/doc_tex/QP_solver/closest_point.gif -text
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QP_solver/doc_tex/QP_solver/closest_point.pdf -text
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QP_solver/doc_tex/QP_solver/first_lp.eps -text
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QP_solver/doc_tex/QP_solver/first_lp.fig -text
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QP_solver/doc_tex/QP_solver/first_lp.gif -text
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@ -1713,6 +1717,7 @@ QP_solver/examples/QP_solver/integer_qp_solver.cin -text
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QP_solver/examples/QP_solver/integer_qp_solver.data -text
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QP_solver/examples/QP_solver/rational_qp_solver.cin -text
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QP_solver/examples/QP_solver/rational_qp_solver.data -text
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QP_solver/examples/QP_solver/shortest_vector.cpp -text
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QP_solver/examples/QP_solver/solve_convex_hull_containment_lp.h -text
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QP_solver/examples/QP_solver/solve_convex_hull_containment_lp2.h -text
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QP_solver/include/CGAL/QP_functions.h -text
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@ -0,0 +1,186 @@
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%!PS-Adobe-2.0 EPSF-2.0
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@ -0,0 +1,55 @@
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#FIG 3.2
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Portrait
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Center
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Inches
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Letter
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100.00
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Single
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2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
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2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
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-6
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1 3 1 1 0 0 47 -1 20 4.000 1 0.0000 6600 3900 75 75 6600 3900 6675 3900
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1 3 1 1 0 0 47 -1 20 4.000 1 0.0000 5700 4800 75 75 5700 4800 5775 4800
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4 0 0 50 -1 1 20 0.0000 4 120 90 5925 4875 q\001
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4 0 0 50 -1 1 20 0.0000 4 120 90 6825 4125 p\001
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@ -53,13 +53,21 @@ the smallest objective function value among all feasible solutions.
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Such a solution may not exist, see Sections
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\ref{sec:QP-infeasible} and \ref{sec:QP-unbounded} below.
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\subsection{Linear and nonnegative programs.}
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\subsection{Linear, nonnegative, and free programs.}
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If $D=0$, the quadratic program is in fact a \emph{linear program},
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and in the case that $l$ is the zero vector and all entries of
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$u$ are $\infty$, the program is said to be \emph{nonnegative}. The
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package offers dedicated solution methods for these special cases,
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see Sections \ref{sec:QP-lp} and \ref{sec:QP-nonnegative} below.
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Often, there are no bounds at all for the variables, i.e.\ all entries
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of $l$ are $-\infty$, and all entries of $u$ are $\infty$. Such a
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program is called \emph{free}. There is no dedicated solution method
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for this case (a free quadratic or linear program is treated like a
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general quadratic or linear program), but the package provides models
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for free programs that save you from specifying the infinite bounds (see
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Section \ref{sec:QP-important_constraints} for an example).
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\subsection{A first example}
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Let's consider the following quadratic program in two variables:
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\[
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@ -379,6 +387,32 @@ contribute to the convex combinations of intermediate points.
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Not surprisingly, only two points contribute to any convex
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combination.
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\section{The Important Constraints}\label{sec:QP-important_constraints}
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If we have a linear or quadratic program with many inequality constraints,
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the ``important'' constraints are typically the ones that are satisfied with
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equality at the optimal solution.
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To demonstrate this, let's consider the problem of finding the point $q$
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in the intersection of halfspaces that is closest to a given point $p$,
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see Figure \ref{fig:QP-closest_point}.
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\begin{figure}[htbp]
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\begin{ccTexOnly}
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\begin{center}
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\includegraphics{QP_solver/closest_point}
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\end{center}
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\end{ccTexOnly}
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\caption{The closest point in the intersection of halfspaces
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\label{fig:QP-closest_point}}
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\begin{ccHtmlOnly}
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<CENTER>
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<IMG BORDER=0 SRC="./closest_point.gif" ALIGN=center ALT="The closest point in the intersection of halfspaces">
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</CENTER>
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\end{ccHtmlOnly}
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\end{figure}
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\section{Working from MPS Files}
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\section{Infeasibility}\label{sec:QP-infeasible}
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@ -386,7 +420,7 @@ combination.
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\section{Unboundedness}\label{sec:QP-unbounded}
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\section{The Important Constraints}
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\section{Solution Certificates}
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// Example: computes point of minimum norm in the intersection of halfspaces
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#include <cassert>
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#include <vector>
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#include <CGAL/Cartesian_d.h>
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#include <CGAL/MP_Float.h>
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#include <boost/iterator/counting_iterator.hpp>
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#include <CGAL/iterator.h>
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#include <CGAL/Kernel_traits.h>
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#include <CGAL/QP_models.h>
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#include <CGAL/QP_functions.h>
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// unary function that maps integer j to iterator constructed from j
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template <class Iterator>
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struct Jth {
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typedef Iterator result_type;
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result_type operator() (int j) const { return Iterator (j); }
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};
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// unary function that maps a hyperplane h to its negative j-th coordinate,
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// for some fixed j
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template <class Hyperplane>
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class Fixed_coordinate {
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public:
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typedef typename CGAL::Kernel_traits<Hyperplane>::Kernel::RT result_type;
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Fixed_coordinate (int index) : j (index) {}
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result_type operator() (const Hyperplane& h) const {return -h[j];}
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private:
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const int j;
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};
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// unary function that maps an integer i to 1 if i=j and 0 otherwise,
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// for some fixed j
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template <class RT>
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class Unit_vector {
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public:
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typedef RT result_type;
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Unit_vector (int index) : j (index) {}
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result_type operator() (int i) { return i==j ? 1 : 0; }
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private:
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const int j;
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};
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// function to solve the QP that computes the minimum-norm point in the
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// intersection of positive halfspaces induced by a set of oriented
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// hyperplanes
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template <class HyperplaneIterator, class ET>
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CGAL::Quadratic_program_solution<ET>
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solve_minimum_norm_point_qp
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(HyperplaneIterator begin, HyperplaneIterator end, const ET& dummy)
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{
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// hyperplane type
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typedef typename HyperplaneIterator::value_type H;
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// input number type
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typedef typename CGAL::Kernel_traits<H>::Kernel::RT RT;
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// iterator type for j-th column of A (j=0,...,d-1) and for b (j=d);
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// transforms iterator for hyperplane to iterator for j-th coordinates
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typedef CGAL::Join_input_iterator_1
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<HyperplaneIterator, Fixed_coordinate<H> > Fixed_coordinate_iterator;
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// iterator type for A; transforms iterator for indices to iterator
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// for corresponding columns of A
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typedef CGAL::Join_input_iterator_1
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<boost::counting_iterator<int>, Jth<Fixed_coordinate_iterator> > A_it;
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// iterator type for b;
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typedef Fixed_coordinate_iterator B_it;
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// iterator type for relations (all are "<=")
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typedef CGAL::Const_oneset_iterator<CGAL::Comparison_result> R_it;
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// iterator type for c (all entries are 0)
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typedef CGAL::Const_oneset_iterator<RT> C_it;
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// iterator type for i-th row of D; transforms iterator for indices
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// to iterator for the i-th unit vector
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typedef CGAL::Join_input_iterator_1
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<boost::counting_iterator<int>, Unit_vector<RT> > Unit_vector_iterator;
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// iterator type for D; transforms iterator for indices to iterator
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// for corresponding rows of D
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typedef CGAL::Join_input_iterator_1
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<boost::counting_iterator<int>, Jth<Unit_vector_iterator> > D_it;
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// determine dimension from first hyperplane
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int d = (*begin).dimension();
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// construct program and solve itx
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return CGAL::solve_quadratic_program
|
||||
(CGAL::make_free_quadratic_program_from_iterators
|
||||
(d, end-begin, A_it(0), B_it(d), R_it(CGAL::SMALLER), D_it(), C_it (0)));
|
||||
}
|
||||
|
||||
int main() {
|
||||
//
|
||||
return 0;
|
||||
|
||||
}
|
||||
|
|
@ -349,6 +349,7 @@ public:
|
|||
d, c, c0)
|
||||
{}
|
||||
};
|
||||
|
||||
// corresponding global function
|
||||
// make_nonnegative_quadratic_program_from_iterators
|
||||
// -------------------------------------------------
|
||||
|
|
@ -400,6 +401,98 @@ public:
|
|||
{}
|
||||
};
|
||||
|
||||
// Free_quadratic_program_from_iterators
|
||||
// -------------------------------------
|
||||
template <
|
||||
typename A_it, // for constraint matrix A (columnwise)
|
||||
typename B_it, // for right-hand side b
|
||||
typename R_it, // for relations (value type Comparison)
|
||||
typename D_it, // for quadratic matrix D (rowwise)
|
||||
typename C_it > // for objective function c
|
||||
class Free_quadratic_program_from_iterators :
|
||||
public Quadratic_program_from_iterators <A_it, B_it, R_it,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
D_it, C_it>
|
||||
{
|
||||
private:
|
||||
typedef Quadratic_program_from_iterators <A_it, B_it, R_it,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
D_it, C_it> Base;
|
||||
typedef typename QP_model_default_iterators<bool*>::It_1d Const_FLU_iterator;
|
||||
typedef typename QP_model_default_iterators<B_it>::It_1d Const_LU_iterator;
|
||||
public:
|
||||
QP_MODEL_ITERATOR_TYPES;
|
||||
Free_quadratic_program_from_iterators (
|
||||
int n, int m, // number of variables / constraints
|
||||
const A_iterator& a,
|
||||
const B_iterator& b,
|
||||
const R_iterator& r,
|
||||
const D_iterator& d,
|
||||
const C_iterator& c,
|
||||
const C_entry& c0 = C_entry(0)
|
||||
)
|
||||
: Base (n, m, a, b, r,
|
||||
Const_FLU_iterator(false), Const_LU_iterator(C_entry(0)),
|
||||
Const_FLU_iterator(false), Const_LU_iterator(C_entry(0)),
|
||||
d, c, c0)
|
||||
{}
|
||||
};
|
||||
// corresponding global function
|
||||
// make_free_quadratic_program_from_iterators
|
||||
// ------------------------------------------
|
||||
template <
|
||||
typename A_it, // for constraint matrix A (columnwise)
|
||||
typename B_it, // for right-hand side b
|
||||
typename R_it, // for relations (value type Comparison)
|
||||
typename D_it, // for quadratic matrix D (rowwise)
|
||||
typename C_it > // for objective function c
|
||||
Free_quadratic_program_from_iterators
|
||||
<A_it, B_it, R_it, D_it, C_it>
|
||||
make_free_quadratic_program_from_iterators (
|
||||
int n, int m,
|
||||
const A_it& a,
|
||||
const B_it& b,
|
||||
const R_it& r,
|
||||
const D_it& d,
|
||||
const C_it& c,
|
||||
typename std::iterator_traits<C_it>::value_type c0 =
|
||||
typename std::iterator_traits<C_it>::value_type(0))
|
||||
{
|
||||
return Free_quadratic_program_from_iterators
|
||||
<A_it, B_it, R_it, D_it, C_it>
|
||||
(n, m, a, b, r, d, c, c0);
|
||||
}
|
||||
|
||||
// Free_Quadratic_program_from_pointers
|
||||
// -------------------------------------------
|
||||
template <typename NT>
|
||||
class Free_quadratic_program_from_pointers :
|
||||
public Free_quadratic_program_from_iterators
|
||||
<NT**, NT*, CGAL::Comparison_result*, NT**, NT*>
|
||||
{
|
||||
private:
|
||||
typedef Free_quadratic_program_from_iterators
|
||||
<NT**, NT*, CGAL::Comparison_result*, NT**, NT*> Base;
|
||||
public:
|
||||
QP_MODEL_ITERATOR_TYPES;
|
||||
Free_quadratic_program_from_pointers (
|
||||
int n, int m, // number of variables / constraints
|
||||
const A_iterator& a,
|
||||
const B_iterator& b,
|
||||
const R_iterator& r,
|
||||
const D_iterator& d,
|
||||
const C_iterator& c,
|
||||
const C_entry& c0 = C_entry(0)
|
||||
)
|
||||
: Base (n, m, a, b, r, d, c, c0)
|
||||
{}
|
||||
};
|
||||
|
||||
|
||||
// Nonnegative_linear_program_from_iterators
|
||||
|
|
@ -492,6 +585,96 @@ public:
|
|||
{}
|
||||
};
|
||||
|
||||
// Free_linear_program_from_iterators
|
||||
// ----------------------------------
|
||||
template <
|
||||
typename A_it, // for constraint matrix A (columnwise)
|
||||
typename B_it, // for right-hand side b
|
||||
typename R_it, // for relations (value type Comparison)
|
||||
typename C_it > // for objective function c
|
||||
class Free_linear_program_from_iterators :
|
||||
public Quadratic_program_from_iterators <A_it, B_it, R_it,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_2d, C_it>
|
||||
{
|
||||
private:
|
||||
typedef Quadratic_program_from_iterators <A_it, B_it, R_it,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
typename QP_model_default_iterators<bool*>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_1d,
|
||||
typename QP_model_default_iterators<B_it>::It_2d, C_it> Base;
|
||||
typedef typename QP_model_default_iterators<bool*>::It_1d Const_FLU_iterator;
|
||||
typedef typename QP_model_default_iterators<B_it>::It_1d Const_LU_iterator;
|
||||
typedef typename QP_model_default_iterators<B_it>::It_2d Const_D_iterator;
|
||||
public:
|
||||
QP_MODEL_ITERATOR_TYPES;
|
||||
Free_linear_program_from_iterators (
|
||||
int n, int m, // number of variables / constraints
|
||||
const A_iterator& a,
|
||||
const B_iterator& b,
|
||||
const R_iterator& r,
|
||||
const C_iterator& c,
|
||||
const C_entry& c0 = C_entry(0)
|
||||
)
|
||||
: Base (n, m, a, b, r,
|
||||
Const_FLU_iterator(false), Const_LU_iterator(C_entry(0)),
|
||||
Const_FLU_iterator(false), Const_LU_iterator(C_entry(0)),
|
||||
Const_D_iterator(C_entry(0)), c, c0)
|
||||
{}
|
||||
};
|
||||
|
||||
// corresponding global function
|
||||
// make_free_linear_program_from_iterators
|
||||
// ---------------------------------------
|
||||
template <
|
||||
typename A_it, // for constraint matrix A (columnwise)
|
||||
typename B_it, // for right-hand side b
|
||||
typename R_it, // for relations (value type Comparison)
|
||||
typename C_it > // for objective function c
|
||||
Nonnegative_linear_program_from_iterators
|
||||
<A_it, B_it, R_it, C_it>
|
||||
make_free_linear_program_from_iterators (
|
||||
int n, int m,
|
||||
const A_it& a,
|
||||
const B_it& b,
|
||||
const R_it& r,
|
||||
const C_it& c,
|
||||
typename std::iterator_traits<C_it>::value_type c0 =
|
||||
typename std::iterator_traits<C_it>::value_type(0))
|
||||
{
|
||||
return Free_linear_program_from_iterators
|
||||
<A_it, B_it, R_it, C_it>
|
||||
(n, m, a, b, r, c, c0);
|
||||
}
|
||||
|
||||
// Free_linear_program_from_pointers
|
||||
// ---------------------------------
|
||||
template <typename NT>
|
||||
class Free_linear_program_from_pointers :
|
||||
public Free_linear_program_from_iterators
|
||||
<NT**, NT*, CGAL::Comparison_result*, NT*>
|
||||
{
|
||||
private:
|
||||
typedef Free_linear_program_from_iterators
|
||||
<NT**, NT*, CGAL::Comparison_result*, NT*> Base;
|
||||
public:
|
||||
QP_MODEL_ITERATOR_TYPES;
|
||||
Free_linear_program_from_pointers (
|
||||
int n, int m, // number of variables / constraints
|
||||
const A_iterator& a,
|
||||
const B_iterator& b,
|
||||
const R_iterator& r,
|
||||
const C_iterator& c,
|
||||
const C_entry& c0 = C_entry(0)
|
||||
)
|
||||
: Base (n, m, a, b, r, c, c0)
|
||||
{}
|
||||
};
|
||||
|
||||
// Quadratic_program_from_mps
|
||||
// --------------------------
|
||||
namespace QP_from_mps_detail {
|
||||
|
|
|
|||
Loading…
Reference in New Issue