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// Copyright (c) 1997-2007 ETH Zurich (Switzerland).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL: https://github.com/CGAL/cgal/blob/v5.1/QP_solver/include/CGAL/QP_solver/QP_solver.h $
// $Id: QP_solver.h 0779373 2020-03-26T13:31:46+01:00 Sébastien Loriot
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s) : Kaspar Fischer
// : Bernd Gaertner <gaertner@inf.ethz.ch>
// : Sven Schoenherr
// : Franz Wessendorp
#ifndef CGAL_QP_SOLVER_H
#define CGAL_QP_SOLVER_H
#include <CGAL/license/QP_solver.h>
#include <CGAL/disable_warnings.h>
#include <CGAL/iterator.h>
#include <CGAL/QP_solver/basic.h>
#include <CGAL/QP_solver/functors.h>
#include <CGAL/QP_options.h>
#include <CGAL/QP_solution.h>
#include <CGAL/QP_solver/QP_basis_inverse.h>
#include <CGAL/QP_solver/QP_pricing_strategy.h>
#include <CGAL/QP_solver/QP_full_exact_pricing.h>
#include <CGAL/QP_solver/QP_partial_exact_pricing.h>
#include <CGAL/QP_solver/QP_full_filtered_pricing.h>
#include <CGAL/QP_solver/QP_partial_filtered_pricing.h>
#include <CGAL/QP_solver/QP_exact_bland_pricing.h>
#include <CGAL/algorithm.h>
#include <CGAL/NT_converter.h>
#include <CGAL/IO/Verbose_ostream.h>
#include <boost/bind.hpp>
#include <boost/function.hpp>
#include <CGAL/boost/iterator/transform_iterator.hpp>
#include <vector>
#include <numeric>
#include <algorithm>
namespace CGAL {
// ==================
// class declarations
// ==================
template < typename Q, typename ET, typename Tags >
class QP_solver;
template <class ET>
class QP_solution;
namespace QP_solver_impl { // namespace for implemenation details
// --------------
// Tags generator
// --------------
template < typename Linear,
typename Nonnegative >
struct QP_tags {
typedef Linear Is_linear;
typedef Nonnegative Is_nonnegative;
};
template < class Q, class Is_linear >
struct D_selector {};
template <class Q>
struct D_selector<Q, Tag_false> // quadratic
{
typedef typename Q::D_iterator D_iterator;
};
template <class Q>
struct D_selector<Q, Tag_true> // linear
{
// dummy type, not used
typedef int** D_iterator;
};
template < class Q, class Is_nonnegative >
struct Bd_selector {};
template < class Q >
struct Bd_selector<Q, Tag_false> // nonstandard form
{
typedef typename Q::FL_iterator FL_iterator;
typedef typename Q::L_iterator L_iterator;
typedef typename Q::FU_iterator FU_iterator;
typedef typename Q::U_iterator U_iterator;
};
template < class Q >
struct Bd_selector<Q, Tag_true> // standard form
{
// dummy types, not used
typedef int* FL_iterator;
typedef int* L_iterator;
typedef int* FU_iterator;
typedef int* U_iterator;
};
// only allow filtered pricing if NT = double
template <typename Q, typename ET, typename Tags, typename NT>
struct Filtered_pricing_strategy_selector
{
typedef QP_full_exact_pricing<Q, ET, Tags> FF;
typedef QP_partial_exact_pricing<Q, ET, Tags> PF;
};
template <typename Q, typename ET, typename Tags>
struct Filtered_pricing_strategy_selector<Q, ET, Tags, double>
{
typedef QP_full_filtered_pricing<Q, ET, Tags> FF;
typedef QP_partial_filtered_pricing<Q, ET, Tags> PF;
};
} // end of namespace for implementation details
// ================
// class interfaces
// ================
template < typename Q, typename ET, typename Tags >
class QP_solver : public QP_solver_base<ET> {
public: // public types
typedef QP_solver<Q, ET, Tags> Self;
typedef QP_solver_base<ET> Base;
// types from the QP
typedef typename Q::A_iterator A_iterator;
typedef typename Q::B_iterator B_iterator;
typedef typename Q::C_iterator C_iterator;
typedef CGAL::Comparison_result Row_type;
typedef typename Q::R_iterator Row_type_iterator;
// the remaining types might not be present in the qp, so the
// following selectors generate dummy types for them
typedef typename QP_solver_impl::
D_selector<Q, typename Tags::Is_linear>::
D_iterator D_iterator;
typedef typename QP_solver_impl::
Bd_selector<Q, typename Tags::Is_nonnegative>::
L_iterator L_iterator;
typedef typename QP_solver_impl::
Bd_selector<Q, typename Tags::Is_nonnegative>::
U_iterator U_iterator;
typedef typename QP_solver_impl::
Bd_selector<Q, typename Tags::Is_nonnegative>::
FL_iterator FL_iterator;
typedef typename QP_solver_impl::
Bd_selector<Q, typename Tags::Is_nonnegative>::
FU_iterator FU_iterator;
// types from the Tags
typedef typename Tags::Is_linear Is_linear;
typedef typename Tags::Is_nonnegative Is_nonnegative;
// friends
template <class Q_, class ET_>
friend bool has_linearly_independent_equations
(const Q_& qp, const ET_& dummy);
private: // private types
// types of original problem:
typedef typename std::iterator_traits<A_iterator>::value_type A_column;
typedef typename std::iterator_traits<D_iterator>::value_type D_row;
typedef typename std::iterator_traits<A_column >::value_type A_entry;
typedef typename std::iterator_traits<B_iterator>::value_type B_entry;
typedef typename std::iterator_traits<C_iterator>::value_type C_entry;
typedef typename std::iterator_traits<D_row >::value_type D_entry;
typedef typename std::iterator_traits<L_iterator>::value_type L_entry;
typedef typename std::iterator_traits<U_iterator>::value_type U_entry;
// slack columns:
//
// The following two types are used to (conceptually) add to the matrix A
// additional columns that model the constraints "x_s>=0" for the slack
// variables x_s. Of course, we do not store the column (which is just
// plus/minus a unit vector), but maintain a pair (int,bool): the first
// entry says in which column the +-1 is and the second entry of the pair
// says whether it is +1 (false) or -1 (true).
typedef std::pair<int,bool> Slack_column;
typedef std::vector<Slack_column> A_slack;
// artificial columns
//
// Artificial columns that are (conceptually) added to the matrix A are
// handled exactly like slack columns (see above).
typedef std::pair<int,bool> Art_column;
typedef std::vector<Art_column> A_art;
// special artificial column:
//
// Also for the special artificial variable we (conceptually) add a column
// to A. This column contains only +-1's (but it may contain several nonzero
// entries).
typedef std::vector<A_entry> S_art;
// auxiliary objective vector (i.e., the objective vector for phase I):
typedef std::vector<C_entry> C_aux;
public: // export some additional types:
typedef typename Base::Indices Indices;
typedef typename Base::Index_mutable_iterator Index_iterator;
typedef typename Base::Index_const_iterator Index_const_iterator;
// For problems in nonstandard form we also export the following type, which
// for an original variable will say whether it sits at is lower, upper, at
// its lower and upper (fixed) bound, or at zero, or whether the variable is
// basic:
enum Bound_index { LOWER, ZERO, UPPER, FIXED, BASIC };
private:
typedef std::vector<Bound_index> Bound_index_values;
typedef typename Bound_index_values::iterator
Bound_index_value_iterator;
typedef typename Bound_index_values::const_iterator
Bound_index_value_const_iterator;
// values (variables' numerators):
typedef std::vector<ET> Values;
typedef typename Values::iterator Value_iterator;
typedef typename Values::const_iterator
Value_const_iterator;
// access values by basic index functor:
typedef CGAL::Value_by_basic_index<Value_const_iterator>
Value_by_basic_index;
// access to original problem by basic variable/constraint index:
typedef QP_vector_accessor<A_column, false, false > A_by_index_accessor;
typedef boost::transform_iterator
< A_by_index_accessor,Index_const_iterator >
A_by_index_iterator;
// todo kf: following can be removed once we have all these (outdated)
// accessors removed:
typedef QP_vector_accessor< B_iterator, false, false >
B_by_index_accessor;
typedef boost::transform_iterator
< B_by_index_accessor, Index_const_iterator >
B_by_index_iterator;
typedef QP_vector_accessor< C_iterator, false, false >
C_by_index_accessor;
typedef boost::transform_iterator
<C_by_index_accessor, Index_const_iterator >
C_by_index_iterator;
typedef QP_matrix_accessor< A_iterator, false, true, false, false>
A_accessor;
typedef boost::function1<typename A_accessor::result_type, int>
A_row_by_index_accessor;
typedef boost::transform_iterator
< A_row_by_index_accessor, Index_iterator >
A_row_by_index_iterator;
// Access to the matrix D sometimes converts to ET, and
// sometimes retruns the original input type
typedef QP_matrix_pairwise_accessor< D_iterator, ET >
D_pairwise_accessor;
typedef boost::transform_iterator
< D_pairwise_accessor, Index_const_iterator>
D_pairwise_iterator;
typedef QP_matrix_pairwise_accessor< D_iterator, D_entry >
D_pairwise_accessor_input_type;
typedef boost::transform_iterator
< D_pairwise_accessor_input_type, Index_const_iterator >
D_pairwise_iterator_input_type;
// access to special artificial column by basic constraint index:
typedef QP_vector_accessor< typename S_art::const_iterator, false, false>
S_by_index_accessor;
typedef boost::transform_iterator
< S_by_index_accessor, Index_iterator >
S_by_index_iterator;
public:
typedef typename A_slack::const_iterator
A_slack_iterator;
typedef typename A_art::const_iterator
A_artificial_iterator;
typedef typename C_aux::const_iterator
C_auxiliary_iterator;
typedef typename Base::Variable_numerator_iterator
Variable_numerator_iterator;
typedef Index_const_iterator Basic_variable_index_iterator;
typedef Value_const_iterator Basic_variable_numerator_iterator;
typedef Index_const_iterator Basic_constraint_index_iterator;
typedef QP_pricing_strategy<Q, ET, Tags> Pricing_strategy;
private:
// compile time tag for symbolic perturbation, should be moved into traits
// class when symbolic perturbation is to be implemented
Tag_false is_perturbed;
// some constants
const ET et0, et1, et2;
// verbose output streams
mutable Verbose_ostream vout; // used for any diagnostic output
mutable Verbose_ostream vout1; // used for some diagnostic output
mutable Verbose_ostream vout2; // used for more diagnostic output
mutable Verbose_ostream vout3; // used for full diagnostic output
mutable Verbose_ostream vout4; // used for output of basis inverse
mutable Verbose_ostream vout5; // used for output of validity tests
// pricing strategy
Pricing_strategy* strategyP;
// given QP
int qp_n; // number of variables
int qp_m; // number of constraints
// min x^T D x + c^T x + c0
A_iterator qp_A; // constraint matrix
B_iterator qp_b; // right-hand-side vector
C_iterator qp_c; // objective vector
C_entry qp_c0; // constant term in objective function
// attention: qp_D represents *twice* the matrix D
D_iterator qp_D; // objective matrix
Row_type_iterator qp_r; // row-types of constraints
FL_iterator qp_fl; // lower bound finiteness vector
L_iterator qp_l; // lower bound vector
FU_iterator qp_fu; // upper bound finiteness vector
U_iterator qp_u; // upper bound vector
A_slack slack_A; // slack part of constraint matrix
// auxiliary problem
A_art art_A; // artificial part of constraint matrix
// Note: in phase I there is an
// additional "fake" column attached
// to this "matrix", see init_basis()
S_art art_s; // special artificial column for slacks
int art_s_i; // art_s_i>=0 -> index of special
// artificial column
// art_s_i==-1 -> no sp. art. col
// art_s_i==-2 -> sp. art. col removed
// after it left basis
int art_basic; // number of basic artificial variables
C_aux aux_c; // objective function for phase I
// initially has the same size as A_art
Indices B_O; // basis (original variables)
// Note: the size of B_O is always
// correct, i.e., equals the number of
// basic original variables, plus (in
// phase I) the number of basic
// artificial variables.
Indices B_S; // basis ( slack variables)
Indices C; // basic constraints ( C = E+S_N )
// Note: the size of C is always
// correct, i.e., corresponds to the
// size of the (conceptual) set
// $E\cup S_N$.
Indices S_B; // nonbasic constraints ( S_B '=' B_S)
QP_basis_inverse<ET,Is_linear>
inv_M_B; // inverse of basis matrix
const ET& d; // reference to `inv_M_B.denominator()'
Values x_B_O; // basic variables (original)
// Note: x_B_O is only enlarged,
// so its size need not be |B|.
Values x_B_S; // basic variables (slack)
Values lambda; // lambda (from KKT conditions)
Bound_index_values x_O_v_i; // bounds value index vector
// the following vectors are updated
// with each update in order to avoid
// evaluating a matrix vector
// multiplication
Values r_C; // r_C = A_{C,N_O}x_{N_O}
// Note: r_C.size() == C.size().
Values r_S_B; // r_S_B = A_{S_B,N_O}x_{N_O}
// The following to variables are initialized (if used at all) in
// transition(). They are not used in case Is_linear or
// Is_nonnegative is set to Tag_true.
Values r_B_O; // r_B_O = 2D_{B_O,N_O}x_{N_O}
Values w; // w = 2D_{O, N_O}x_{N_O}
int m_phase; // phase of the Simplex method
Quadratic_program_status m_status; // status of last pivot step
int m_pivots; // number of pivot steps
bool is_phaseI; // flag indicating phase I
bool is_phaseII;// flag indicating phase II
bool is_RTS_transition; // flag indicating transition
// from Ratio Test Step1 to Ratio
// Test Step2
const bool is_LP; // flag indicating a linear program
const bool is_QP; // flag indicating a quadratic program
// the following flag indicates whether the program is in equational form
// AND still has all its equations; this is given in phase I for any
// program in equational form, but it may change if redundant constraints
// get removed from the basis. If no_ineq == true, the program is treated
// in a more efficient manner, since in that case we need no bookkeeping
// for basic constraints
bool no_ineq;
bool has_ineq; // !no_ineq
const bool is_nonnegative; // standard form, from Tag
// additional variables
int l; // minimum of 'qp_n+e+1' and 'qp_m'
// Note: this is an upper bound for
// the size of the reduced basis in
// phase I (in phase II, the reduced
// basis size can be arbitrarily
// large)
int e; // number of equality constraints
// Given a variable number i, in_B[i] is -1 iff x_i is not in the current
// basis. If the number in_B[i] is >=0, it is the basis heading of x_i.
Indices in_B; // variable in basis, -1 if non-basic
// Given a number i in {0,...,qp_m-1} of a constraint,
Indices in_C; // constraint in basis, -1 if non-basic
// Note: in_C is only maintained if
// there are inequality constraints.
Values b_C; // exact version of `qp_b'
// restricted to basic constraints C
Values minus_c_B; // exact version of `-qp_c'
// restricted to basic variables B_O
// Note: minus_c_B is only enlarged,
// so its size need not be |B|.
Values A_Cj; // exact version of j-th column of A
// restricted to basic constraints C
Values two_D_Bj; // exact version of twice the j-th
// column of D restricted to B_O
// Note: tmp_x_2 is only enlarged,
// so its size need not be |B|.
int j; // index of entering variable `x_j'
int i; // index of leaving variable `x_i'
ET x_i; // numerator of leaving variable `x_i'
ET q_i; // corresponding `q_i'
int direction; // indicates whether the current
// entering variable x_j is increased
// or decreased
Bound_index ratio_test_bound_index; // indicates for leaving
// original variables which bound
// was hit with upper bounding
ET mu; // numerator of `t_j'
ET nu; // denominator of `t_j'
Values q_lambda; // length dependent on C
Values q_x_O; // used in the ratio test & update
// Note: q_x_O is only enlarged,
// so its size need not be |B|.
Values q_x_S; //
Values tmp_l; // temporary vector of size l
Values tmp_x; // temporary vector of s. >= B_O.size()
// Note: tmp_x is only enlarged,
// so its size need not be |B|.
Values tmp_l_2; // temporary vector of size l
Values tmp_x_2; // temporary vector of s. >= B_O.size()
// Note: tmp_x_2 is only enlarged,
// so its size need not be |B|.
// Diagnostics
struct Diagnostics {
bool redundant_equations;
};
Diagnostics diagnostics;
public:
/*
* Note: Some member functions below are suffixed with '_'.
* They are member templates and their declaration is "hidden",
* because they are also implemented in the class interface.
* This is a workaround for M$-VC++, which otherwise fails to
* instantiate them correctly.
*/
// creation & initialization
// -------------------------
// creation
QP_solver(const Q& qp,
const Quadratic_program_options& options =
Quadratic_program_options());
virtual ~QP_solver()
{
if (strategyP != static_cast<Pricing_strategy*>(0))
delete strategyP;
}
private:
// set-up of QP
void set( const Q& qp);
void set_D (const Q& qp, Tag_true is_linear);
void set_D (const Q& qp, Tag_false is_linear);
// set-up of explicit bounds
void set_explicit_bounds(const Q& qp);
void set_explicit_bounds(const Q& qp, Tag_true /*is_nonnegative*/);
void set_explicit_bounds(const Q& qp, Tag_false /*is_nonnegative*/);
// initialization (of phase I)
void init( );
// initialization (of phase II)
/*
template < class InputIterator >
void init( InputIterator basic_variables_first,
InputIterator basic_variables_beyond);
*/
// operations
// ----------
// pivot step
Quadratic_program_status pivot( )
{ CGAL_qpe_assertion( phase() > 0);
CGAL_qpe_assertion( phase() < 3);
pivot_step();
return status(); }
// solve QP
Quadratic_program_status solve( )
{ CGAL_qpe_assertion( phase() > 0);
while ( phase() < 3) { pivot_step(); }
return status(); }
public:
// access
// ------
// access to QP
int number_of_variables ( ) const { return qp_n; }
int number_of_constraints( ) const { return qp_m; }
A_iterator a_begin( ) const { return qp_A; }
A_iterator a_end ( ) const { return qp_A+qp_n; }
B_iterator b_begin( ) const { return qp_b; }
B_iterator b_end ( ) const { return qp_b+qp_m; }
C_iterator c_begin( ) const { return qp_c; }
C_iterator c_end ( ) const { return qp_c+qp_n; }
C_entry c_0 ( ) const { return qp_c0;}
D_iterator d_begin( ) const { return qp_D; }
D_iterator d_end ( ) const { return qp_D+qp_n; }
Row_type_iterator row_type_begin( ) const { return qp_r; }
Row_type_iterator row_type_end ( ) const { return qp_r+qp_m; }
// access to current status
int phase ( ) const { return m_phase; }
Quadratic_program_status status ( ) const { return m_status; }
int iterations( ) const { return m_pivots; }
// access to common denominator
const ET& variables_common_denominator( ) const
{
CGAL_qpe_assertion (d > 0);
return d;
}
// access to current solution
ET solution_numerator( ) const;
// access to current solution
ET solution_denominator( ) const { return et2*d*d; }
// access to original variables
int number_of_original_variables( ) const { return qp_n; }
// access to slack variables
int number_of_slack_variables( ) const { return static_cast<int>(slack_A.size()); }
// access to artificial variables
int number_of_artificial_variables( ) const { return static_cast<int>(art_A.size()); }
C_auxiliary_iterator
c_auxiliary_value_iterator_begin( ) const { return aux_c.begin(); }
C_auxiliary_iterator
c_auxiliary_value_iterator_end( ) const {return aux_c.end(); }
// access to basic variables
int number_of_basic_variables( ) const { return static_cast<int>(B_O.size()+B_S.size()); }
int number_of_basic_original_variables( ) const { return static_cast<int>(B_O.size()); }
int number_of_basic_slack_variables( ) const { return static_cast<int>(B_S.size()); }
Basic_variable_index_iterator
basic_original_variable_indices_begin( ) const { return B_O.begin(); }
Basic_variable_index_iterator
basic_original_variable_indices_end ( ) const { return B_O.end(); }
Basic_variable_numerator_iterator
basic_original_variables_numerator_begin( ) const { return x_B_O.begin(); }
Basic_variable_numerator_iterator
basic_original_variables_numerator_end ( ) const { return x_B_O.begin()
+ B_O.size(); }
public: // only the pricing strategies (including user-defined ones
// need access to this) -- make them friends?
// access to working variables
int number_of_working_variables( ) const { return static_cast<int>(in_B.size()); }
bool is_basic( int j) const
{
CGAL_qpe_assertion(j >= 0);
CGAL_qpe_assertion(j < number_of_working_variables());
return (in_B[ j] >= 0);
}
bool is_original(int j) const
{
CGAL_qpe_assertion(j >= 0);
CGAL_qpe_assertion(j < number_of_working_variables());
return (j < qp_n);
}
bool phaseI( ) const {return is_phaseI;}
bool is_artificial(int k) const;
int get_l() const;
// Returns w[j] for an original variable x_j.
ET w_j_numerator(int j) const
{
CGAL_qpe_assertion((0 <= j) && (j < qp_n) && is_phaseII);
return w[j];
}
Bound_index nonbasic_original_variable_bound_index(int i) const
// Returns on which bound the nonbasic variable x_i is currently
// sitting:
//
// - LOWER: the variable is sitting on its lower bound.
// - UPPER: the variable is sitting on its upper bound.
// - FIXED: the variable is sitting on its lower and upper bound.
// - ZERO: the variable has value zero and is sitting on its lower
// bound, its upper bound, or betweeen the two bounds.
//
// Note: in the latter case you can call state_of_zero_nonbasic_variable()
// to find out which bound is active, if any.
{
CGAL_assertion(!check_tag(Is_nonnegative()) &&
!is_basic(i) && i < qp_n);
if (x_O_v_i[i] == BASIC) {
CGAL_qpe_assertion(false);
}
return x_O_v_i[i];
};
int state_of_zero_nonbasic_variable(int i) const
// Returns -1 if the original variable x_i equals its lower bound,
// 0 if it lies strictly between its lower and upper bound, and 1 if
// it coincides with its upper bound.
//
// See also the documentation of nonbasic_original_variable_bound_index()
// above.
{
CGAL_assertion(!check_tag(Is_nonnegative()) &&
!is_basic(i) && i < qp_n && x_O_v_i[i] == ZERO);
if (*(qp_fl+i) && CGAL::is_zero(*(qp_l+i)))
return -1;
if (*(qp_fu+i) && CGAL::is_zero(*(qp_u+i)))
return 1;
return 0;
}
private:
// miscellaneous
// -------------
// setting the pricing strategy:
void set_pricing_strategy ( Quadratic_program_pricing_strategy strategy);
// diagnostic output
void set_verbosity( int verbose = 0, std::ostream& stream = std::cout);
public:
// access to indices of basic constraints
int number_of_basic_constraints( ) const { return static_cast<int>(C.size()); }
Basic_constraint_index_iterator
basic_constraint_indices_begin( ) const { return C.begin(); }
Basic_constraint_index_iterator
basic_constraint_indices_end ( ) const { return C.end(); }
// helper functions
template < class RndAccIt1, class RndAccIt2, class NT >
NT mu_j_( int j, RndAccIt1 lambda_it, RndAccIt2 x_it, const NT& dd) const;
ET dual_variable( int i)
{
for ( int j = 0; j < qp_m; ++j) {
tmp_x[ j] = inv_M_B.entry( j, i);
}
return std::inner_product( tmp_x.begin(), tmp_x.begin()+qp_m,
minus_c_B.begin(), et0);
}
public:
// public access to compressed lambda (used in filtered base)
Value_const_iterator get_lambda_begin() const
{
return lambda.begin();
}
Value_const_iterator get_lambda_end() const
{
return lambda.begin() + C.size();
}
private:
// private member functions
// ------------------------
// initialization
void init_basis( );
void init_basis__slack_variables( int s_i, Tag_true has_no_inequalities);
void init_basis__slack_variables( int s_i, Tag_false has_no_inequalities);
void init_basis__slack_variables( int s_i, bool has_no_inequalities) {
if (has_no_inequalities)
init_basis__slack_variables (s_i, Tag_true());
else
init_basis__slack_variables (s_i, Tag_false());
}
void init_basis__constraints ( int s_i, Tag_true has_no_inequalities);
void init_basis__constraints ( int s_i, Tag_false has_no_inequalities);
void init_basis__constraints ( int s_i, bool has_no_inequalities) {
if (has_no_inequalities)
init_basis__constraints (s_i, Tag_true());
else
init_basis__constraints (s_i, Tag_false());
}
void init_x_O_v_i();
void init_r_C(Tag_true /*is_nonnegative*/);
void init_r_C(Tag_false /*is_nonnegative*/);
void init_r_S_B(Tag_true /*is_nonnegative*/);
void init_r_S_B(Tag_false /*is_nonnegative*/);
void init_r_B_O();
void init_w();
void init_solution( );
void init_solution__b_C( Tag_true has_no_inequalities);
void init_solution__b_C( Tag_false has_no_inequalities);
void init_solution__b_C( bool has_no_inequalities) {
if (has_no_inequalities)
init_solution__b_C (Tag_true());
else
init_solution__b_C (Tag_false());
}
void init_additional_data_members( );
// function needed for set up of auxiliary problem for symbolic perturbation
int signed_leading_exponent( int row);
// This is a variant of set_up_auxiliary_problem for symbolic perturbation
// for the perturbed case
void set_up_auxiliary_problemI( Tag_true is_perturbed);
void set_up_auxiliary_problem();
// transition (to phase II)
void transition( );
void transition( Tag_true is_linear);
void transition( Tag_false is_linear);
// pivot step
void pivot_step( );
// pricing
void pricing( );
template < class NT, class It >
void mu_j__linear_part_( NT& mu_j, int j, It lambda_it,
Tag_true has_no_inequalities) const;
template < class NT, class It >
void mu_j__linear_part_( NT& mu_j, int j, It lambda_it,
Tag_false has_no_inequalities) const;
template < class NT, class It >
void mu_j__linear_part_( NT& mu_j, int j, It lambda_it,
bool has_no_inequalities) const {
if (has_no_inequalities)
mu_j__linear_part_ (mu_j, j, lambda_it, Tag_true());
else
mu_j__linear_part_ (mu_j, j, lambda_it, Tag_false());
}
// template < class NT, class It >
// void mu_j__quadratic_part_( NT& mu_j, int j, It x_it,
// Tag_true is_linear) const;
// template < class NT, class It >
// void mu_j__quadratic_part_( NT& mu_j, int j, It x_it,
// Tag_false is_linear) const;
// template < class NT, class It >
// void mu_j__quadratic_part_( NT& mu_j, int j, It x_it,
// Tag_false is_linear,
// Tag_true is_symmetric) const;
// template < class NT, class It >
// void mu_j__quadratic_part_( NT& mu_j, int j, It x_it,
// Tag_false is_linear,
// Tag_false is_symmetric) const;
template < class NT, class It >
void mu_j__slack_or_artificial_( NT& mu_j, int j, It lambda_it,
const NT& dd,
Tag_true has_no_inequalities) const;
template < class NT, class It >
void mu_j__slack_or_artificial_( NT& mu_j, int j, It lambda_it,
const NT& dd,
Tag_false has_no_inequalities) const;
template < class NT, class It >
void mu_j__slack_or_artificial_( NT& mu_j, int j, It lambda_it,
const NT& dd,
bool has_no_inequalities) const {
if (has_no_inequalities)
mu_j__slack_or_artificial_ (mu_j, j, lambda_it, dd, Tag_true());
else
mu_j__slack_or_artificial_ (mu_j, j, lambda_it, dd, Tag_false());
}
// ratio test
void ratio_test_init( );
void ratio_test_init__A_Cj( Value_iterator A_Cj_it, int j,
Tag_true has_no_inequalities);
void ratio_test_init__A_Cj( Value_iterator A_Cj_it, int j,
Tag_false has_no_inequalities);
void ratio_test_init__A_Cj( Value_iterator A_Cj_it, int j,
bool has_no_inequalities) {
if (has_no_inequalities)
ratio_test_init__A_Cj (A_Cj_it, j, Tag_true());
else
ratio_test_init__A_Cj (A_Cj_it, j, Tag_false());
}
void ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j,
Tag_true is_linear);
void ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j,
Tag_false is_linear);
void ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j,
Tag_false is_linear,
Tag_true has_no_inequalities);
void ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j,
Tag_false is_linear,
Tag_false has_no_inequalities);
void ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j,
Tag_false is_linear,
bool has_no_inequalities) {
if (has_no_inequalities)
ratio_test_init__2_D_Bj( two_D_Bj_it, j, is_linear, Tag_true());
else
ratio_test_init__2_D_Bj( two_D_Bj_it, j, is_linear, Tag_false());
}
void ratio_test_1( );
void ratio_test_1__q_x_O( Tag_true is_linear);
void ratio_test_1__q_x_O( Tag_false is_linear);
void ratio_test_1__q_x_S( Tag_true has_no_inequalities);
void ratio_test_1__q_x_S( Tag_false has_no_inequalities);
void ratio_test_1__q_x_S( bool has_no_inequalities) {
if (has_no_inequalities)
ratio_test_1__q_x_S (Tag_true());
else
ratio_test_1__q_x_S (Tag_false());
}
void ratio_test_1__t_min_j(Tag_true /*is_nonnegative*/);
void ratio_test_1__t_min_j(Tag_false /*is_nonnegative*/);
void ratio_test_1__t_i( Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it,
Tag_true no_check);
void ratio_test_1__t_i( Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it,
Tag_false no_check);
// replaces the above two functions
void ratio_test_1__t_min_B(Tag_true has_no_inequalities );
void ratio_test_1__t_min_B(Tag_false has_no_inequalities );
void ratio_test_1__t_min_B(bool has_no_inequalities ) {
if (has_no_inequalities)
ratio_test_1__t_min_B (Tag_true());
else
ratio_test_1__t_min_B (Tag_false());
}
void ratio_test_1_B_O__t_i(Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it,
Tag_true /*is_nonnegative*/);
void ratio_test_1_B_O__t_i(Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it,
Tag_false /*is_nonnegative*/);
void ratio_test_1_B_S__t_i(Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it,
Tag_true /*is_nonnegative*/);
void ratio_test_1_B_S__t_i(Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it,
Tag_false /*is_nonnegative*/);
void test_implicit_bounds_dir_pos(int k, const ET& x_k, const ET& q_k,
int& i_min, ET& d_min, ET& q_min);
void test_implicit_bounds_dir_neg(int k, const ET& x_k, const ET& q_k,
int& i_min, ET& d_min, ET& q_min);
void test_explicit_bounds_dir_pos(int k, const ET& x_k, const ET& q_k,
int& i_min, ET& d_min, ET& q_min);
void test_explicit_bounds_dir_neg(int k, const ET& x_k, const ET& q_k,
int& i_min, ET& d_min, ET& q_min);
void test_mixed_bounds_dir_pos(int k, const ET& x_k, const ET& q_k,
int& i_min, ET& d_min, ET& q_min);
void test_mixed_bounds_dir_neg(int k, const ET& x_k, const ET& q_k,
int& i_min, ET& d_min, ET& q_min);
void ratio_test_1__t_j( Tag_true is_linear);
void ratio_test_1__t_j( Tag_false is_linear);
void ratio_test_2( Tag_true is_linear);
void ratio_test_2( Tag_false is_linear);
void ratio_test_2__p( Tag_true has_no_inequalities);
void ratio_test_2__p( Tag_false has_no_inequalities);
void ratio_test_2__p( bool has_no_inequalities) {
if (has_no_inequalities)
ratio_test_2__p (Tag_true());
else
ratio_test_2__p (Tag_false());
}
// update
void update_1( );
void update_1( Tag_true is_linear);
void update_1( Tag_false is_linear);
void update_2( Tag_true is_linear);
void update_2( Tag_false is_linear);
void replace_variable( );
void replace_variable( Tag_true has_no_inequalities);
void replace_variable( Tag_false has_no_inequalities);
void replace_variable( bool has_no_inequalities) {
if (has_no_inequalities)
replace_variable (Tag_true());
else
replace_variable (Tag_false());
}
void replace_variable_original_original( );
// update of the vector r
void replace_variable_original_original_upd_r(Tag_true
/*is_nonnegative*/);
void replace_variable_original_original_upd_r(Tag_false
/*is_nonnegative*/);
void replace_variable_original_slack( );
// update of the vector r
void replace_variable_original_slack_upd_r(Tag_true /*is_nonnegative*/);
void replace_variable_original_slack_upd_r(Tag_false /*is_nonnegative*/);
void replace_variable_slack_original( );
// update of the vector r
void replace_variable_slack_original_upd_r(Tag_true /*is_nonnegative*/);
void replace_variable_slack_original_upd_r(Tag_false /*is_nonnegative*/);
void replace_variable_slack_slack( );
// update of the vector r
void replace_variable_slack_slack_upd_r(Tag_true /*is_nonnegative*/);
void replace_variable_slack_slack_upd_r(Tag_false /*is_nonnegative*/);
void remove_artificial_variable_and_constraint( );
// update of the vector r
void remove_artificial_variable_and_constraint_upd_r(Tag_true
/*is_nonnegative*/);
void remove_artificial_variable_and_constraint_upd_r(Tag_false
/*is_nonnegative*/);
void expel_artificial_variables_from_basis( );
// update that occurs only with upper bounding in ratio test step 1
void enter_and_leave_variable( );
void enter_variable( );
// update of the vectors w and r
void enter_variable_original_upd_w_r(Tag_true /*is_nonnegative*/);
void enter_variable_original_upd_w_r(Tag_false /*is_nonnegative*/);
void enter_variable_slack_upd_w_r(Tag_true /*is_nonnegative*/);
void enter_variable_slack_upd_w_r(Tag_false /*is_nonnegative*/);
void leave_variable( );
// update of the vectors w and r
void leave_variable_original_upd_w_r(Tag_true /*is_nonnegative*/);
void leave_variable_original_upd_w_r(Tag_false /*is_nonnegative*/);
void leave_variable_slack_upd_w_r(Tag_true /*is_nonnegative*/);
void leave_variable_slack_upd_w_r(Tag_false /*is_nonnegative*/);
void z_replace_variable( );
void z_replace_variable( Tag_true has_no_inequalities);
void z_replace_variable( Tag_false has_no_inequalities);
void z_replace_variable( bool has_no_inequalities) {
if (has_no_inequalities)
z_replace_variable (Tag_true());
else
z_replace_variable (Tag_false());
}
void z_replace_variable_original_by_original( );
// update of the vectors w and r
void z_replace_variable_original_by_original_upd_w_r(Tag_true
/*is_nonnegative*/);
void z_replace_variable_original_by_original_upd_w_r(Tag_false
/*is_nonnegative*/);
void z_replace_variable_original_by_slack( );
// update of the vectors w and r
void z_replace_variable_original_by_slack_upd_w_r(Tag_true
/*is_nonnegative*/);
void z_replace_variable_original_by_slack_upd_w_r(Tag_false
/*is_nonnegative*/);
void z_replace_variable_slack_by_original( );
// update of the vectors w and r
void z_replace_variable_slack_by_original_upd_w_r(Tag_true
/*is_nonnegative*/);
void z_replace_variable_slack_by_original_upd_w_r(Tag_false
/*is_nonnegative*/);
void z_replace_variable_slack_by_slack( );
// update of the vectors w and r
void z_replace_variable_slack_by_slack_upd_w_r(Tag_true
/*is_nonnegative*/);
void z_replace_variable_slack_by_slack_upd_w_r(Tag_false
/*is_nonnegative*/);
// update of the parts r_C and r_S_B
void update_r_C_r_S_B__j(ET& x_j);
void update_r_C_r_S_B__j_i(ET& x_j, ET& x_i);
void update_r_C_r_S_B__i(ET& x_i);
// update of w and r_B_O
void update_w_r_B_O__j(ET& x_j);
void update_w_r_B_O__j_i(ET& x_j, ET& x_i);
void update_w_r_B_O__i(ET& x_i);
bool basis_matrix_stays_regular( );
// current solution
void compute_solution(Tag_true /*is_nonnegative*/);
void compute_solution(Tag_false /*is_nonnegative*/);
void compute__x_B_S( Tag_false has_no_inequalities,
Tag_false /*is_nonnegative*/);
void compute__x_B_S( Tag_false has_no_inequalities,
Tag_true /*is_nonnegative*/);
void compute__x_B_S( Tag_true has_no_inequalities,
Tag_false /*is_nonnegative*/);
void compute__x_B_S( Tag_true has_no_inequalities,
Tag_true /*is_nonnegative*/);
void compute__x_B_S( bool has_no_inequalities,
Tag_true is_nonnegative) {
if (has_no_inequalities)
compute__x_B_S (Tag_true(), is_nonnegative);
else
compute__x_B_S (Tag_false(), is_nonnegative);
}
void compute__x_B_S( bool has_no_inequalities,
Tag_false is_nonnegative) {
if (has_no_inequalities)
compute__x_B_S (Tag_true(), is_nonnegative);
else
compute__x_B_S (Tag_false(), is_nonnegative);
}
void multiply__A_S_BxB_O( Value_iterator in, Value_iterator out) const;
ET multiply__A_ixO(int row) const;
void multiply__A_CxN_O(Value_iterator out) const;
bool check_r_C(Tag_true /*is_nonnegative*/) const;
bool check_r_C(Tag_false /*is_nonnegative*/) const;
void multiply__A_S_BxN_O(Value_iterator out) const;
bool check_r_S_B(Tag_true /*is_nonnegative*/) const;
bool check_r_S_B(Tag_false /*is_nonnegative*/) const;
void multiply__2D_B_OxN_O(Value_iterator out) const;
bool check_r_B_O(Tag_true /*is_nonnegative*/) const;
bool check_r_B_O(Tag_false /*is_nonnegative*/) const;
void multiply__2D_OxN_O(Value_iterator out) const;
bool check_w(Tag_true /*is_nonnegative*/) const;
bool check_w(Tag_false /*is_nonnegative*/) const;
// utility routines for QP's in nonstandard form:
ET original_variable_value_under_bounds(int i) const;
ET nonbasic_original_variable_value (int i) const;
public:
// for original variables
ET variable_numerator_value(int i) const;
ET unbounded_direction_value(int i) const;
ET lambda_numerator(int i) const
{
// we use the vector lambda which conforms to C (basic constraints)
CGAL_qpe_assertion (i >= 0);
CGAL_qpe_assertion (i <= qp_m);
if (no_ineq)
return lambda[i];
else {
int k = in_C[i]; // position of i in C
if (k != -1)
return lambda[k];
else
return et0;
}
}
private:
// check basis inverse
bool check_basis_inverse( );
bool check_basis_inverse( Tag_true is_linear);
bool check_basis_inverse( Tag_false is_linear);
// diagnostic output
void print_program ( ) const;
void print_basis ( ) const;
void print_solution( ) const;
void print_ratio_1_original(int k, const ET& x_k, const ET& q_k);
void print_ratio_1_slack(int k, const ET& x_k, const ET& q_k);
const char* variable_type( int k) const;
// ensure container size
template <class Container>
void ensure_size(Container& c, typename Container::size_type desired_size) {
typedef typename Container::value_type Value_type;
for (typename Container::size_type i=c.size(); i < desired_size; ++i) {
c.push_back(Value_type());
}
}
private:
private: // (inefficient) access to bounds of variables:
// Given an index of an original or slack variable, returns whether
// or not the variable has a finite lower bound.
bool has_finite_lower_bound(int i) const;
// Given an index of an original or slack variable, returns whether
// or not the variable has a finite upper bound.
bool has_finite_upper_bound(int i) const;
// Given an index of an original or slack variable, returns its
// lower bound.
ET lower_bound(int i) const;
// Given an index of an original variable, returns its upper bound.
ET upper_bound(int i) const;
struct Bnd { // (inefficient) utility class representing a possibly
// infinite bound
enum Kind { MINUS_INF=-1, FINITE=0, PLUS_INF=1 };
const Kind kind; // whether the bound is finite or not
const ET value; // bound's value in case it is finite
Bnd(bool is_upper, bool is_finite, const ET& value)
: kind(is_upper? (is_finite? FINITE : PLUS_INF) :
(is_finite? FINITE : MINUS_INF)),
value(value) {}
Bnd(Kind kind, const ET& value) : kind(kind), value(value) {}
bool operator==(const ET& v) const { return kind == FINITE && value == v; }
bool operator==(const Bnd& b) const {
return kind == b.kind && (kind != FINITE || value == b.value);
}
bool operator!=(const Bnd& b) const { return !(*this == b); }
bool operator<(const ET& v) const { return kind == FINITE && value < v; }
bool operator<(const Bnd& b) const {
return kind < b.kind ||
(kind == b.kind && kind == FINITE && value < b.value);
}
bool operator<=(const Bnd& b) const { return *this < b || *this == b; }
bool operator>(const ET& v) const { return kind == FINITE && value > v; }
bool operator>(const Bnd& b) const { return !(*this <= b); }
bool operator>=(const Bnd& b) const { return !(*this < b); }
Bnd operator*(const ET& f) const { return Bnd(kind, value*f); }
};
// Given an index of an original, slack, or artificial variable,
// return its lower bound.
Bnd lower_bnd(int i) const;
// Given an index of an original, slack, or artificial variable,
// return its upper bound.
Bnd upper_bnd(int i) const;
private:
bool is_value_correct() const;
// ----------------------------------------------------------------------------
// ===============================
// class implementation (template)
// ===============================
public:
// pricing
// -------
// The solver provides three methods to compute mu_j; the first
// two below take additional information (which the pricing
// strategy either provides in exact- or NT-form), and the third
// simply does the exact computation. (Note: internally, we use
// the third version, too, see ratio_test_1__t_j().)
// computation of mu_j with standard form
template < class RndAccIt1, class RndAccIt2, class NT >
NT
mu_j( int j, RndAccIt1 lambda_it, RndAccIt2 x_it, const NT& dd) const
{
NT mu_j;
if ( j < qp_n) { // original variable
// [c_j +] A_Cj^T * lambda_C
mu_j = ( is_phaseI ? NT( 0) : dd * NT(*(qp_c+ j)));
mu_j__linear_part( mu_j, j, lambda_it, no_ineq);
// ... + 2 D_Bj^T * x_B
mu_j__quadratic_part( mu_j, j, x_it, Is_linear());
} else { // slack or artificial
mu_j__slack_or_artificial( mu_j, j, lambda_it, dd,
no_ineq);
}
return mu_j;
}
// computation of mu_j with upper bounding
template < class RndAccIt1, class RndAccIt2, class NT >
NT
mu_j( int j, RndAccIt1 lambda_it, RndAccIt2 x_it, const NT& w_j,
const NT& dd) const
{
NT mu_j;
if ( j < qp_n) { // original variable
// [c_j +] A_Cj^T * lambda_C
mu_j = ( is_phaseI ? NT( 0) : dd * NT(*(qp_c+ j)));
mu_j__linear_part( mu_j, j, lambda_it, no_ineq);
// ... + 2 D_Bj^T * x_B + 2 D_Nj x_N
mu_j__quadratic_part( mu_j, j, x_it, w_j, dd, Is_linear());
} else { // slack or artificial
mu_j__slack_or_artificial( mu_j, j, lambda_it, dd,
no_ineq);
}
return mu_j;
}
// computation of mu_j (exact, both for upper bounding and standard form)
ET
mu_j( int j) const
{
CGAL_qpe_assertion(!is_basic(j));
if (!check_tag(Is_nonnegative()) &&
!check_tag(Is_linear()) &&
!is_phaseI && is_original(j)) {
return mu_j(j,
lambda.begin(),
basic_original_variables_numerator_begin(),
w_j_numerator(j),
variables_common_denominator());
} else {
return mu_j(j,
lambda.begin(),
basic_original_variables_numerator_begin(),
variables_common_denominator());
}
}
private:
// pricing (private helper functions)
// ----------------------------------
template < class NT, class It > inline // no ineq.
void
mu_j__linear_part( NT& mu_j, int j, It lambda_it, Tag_true) const
{
mu_j += inv_M_B.inner_product_l( lambda_it, *(qp_A+ j));
}
template < class NT, class It > inline // has ineq.
void
mu_j__linear_part( NT& mu_j, int j, It lambda_it, Tag_false) const
{
mu_j += inv_M_B.inner_product_l
( lambda_it,
A_by_index_iterator( C.begin(),
A_by_index_accessor( *(qp_A + j))));
}
template < class NT, class It > inline
void
mu_j__linear_part( NT& mu_j, int j, It lambda_it,
bool has_no_inequalities) const {
if (has_no_inequalities)
mu_j__linear_part (mu_j, j, lambda_it, Tag_true());
else
mu_j__linear_part (mu_j, j, lambda_it, Tag_false());
}
template < class NT, class It > inline // LP case, standard form
void
mu_j__quadratic_part( NT&, int, It, Tag_true) const
{
// nop
}
template < class NT, class It > inline // LP case, upper bounded
void
mu_j__quadratic_part( NT&, int, It, const NT& /*w_j*/, const NT& /*dd*/,
Tag_true) const
{
// nop
}
template < class NT, class It > inline // QP case, standard form
void
mu_j__quadratic_part( NT& mu_j, int j, It x_it, Tag_false) const
{
if ( is_phaseII) {
// 2 D_Bj^T * x_B
mu_j += inv_M_B.inner_product_x
( x_it,
D_pairwise_iterator_input_type( B_O.begin(),
D_pairwise_accessor_input_type(qp_D, j)));
}
}
template < class NT, class It > inline // QP case, upper bounded
void
mu_j__quadratic_part( NT& mu_j, int j, It x_it, const NT& w_j,
const NT& dd, Tag_false) const
{
if ( is_phaseII) {
mu_j += dd * w_j;
// 2 D_Bj^T * x_B
mu_j += inv_M_B.inner_product_x
( x_it,
D_pairwise_iterator_input_type( B_O.begin(),
D_pairwise_accessor_input_type(qp_D, j)));
}
}
template < class NT, class It > inline // no ineq.
void
mu_j__slack_or_artificial( NT& mu_j, int j, It lambda_it, const NT& dd, Tag_true) const
{
j -= qp_n;
// artificial variable
// A_j^T * lambda
mu_j = lambda_it[ j];
if ( art_A[ j].second) mu_j = -mu_j;
// c_j + ...
mu_j += dd*NT(aux_c[ j]);
}
template < class NT, class It > inline // has ineq.
void
mu_j__slack_or_artificial( NT& mu_j, int j, It lambda_it, const NT& dd, Tag_false) const
{
j -= qp_n;
if ( j < static_cast<int>(slack_A.size())) { // slack variable
// A_Cj^T * lambda_C
mu_j = lambda_it[ in_C[ slack_A[ j].first]];
if ( slack_A[ j].second) mu_j = -mu_j;
} else { // artificial variable
j -= static_cast<int>(slack_A.size());
// A_Cj^T * lambda_C
mu_j = lambda_it[ in_C[ art_A[ j].first]];
if ( art_A[ j].second) mu_j = -mu_j;
// c_j + ...
mu_j += dd*NT(aux_c[ j]);
}
}
template < class NT, class It > inline
void
mu_j__slack_or_artificial( NT& mu_j, int j, It lambda_it,
const NT& dd, bool has_no_inequalities) const {
if (has_no_inequalities)
mu_j__slack_or_artificial (mu_j, j, lambda_it, dd, Tag_true());
else
mu_j__slack_or_artificial (mu_j, j, lambda_it, dd, Tag_false());
}
};
// ----------------------------------------------------------------------------
// =============================
// class implementation (inline)
// =============================
// initialization
// --------------
// transition
// ----------
template < class Q, typename ET, typename Tags > inline // QP case
void QP_solver<Q, ET, Tags>::
transition( Tag_false)
{
typedef Creator_2< D_iterator, int,
D_pairwise_accessor > D_transition_creator_accessor;
typedef Creator_2< Index_iterator, D_pairwise_accessor,
D_pairwise_iterator > D_transition_creator_iterator;
// initialization of vector w and vector r_B_O:
if (!check_tag(Is_nonnegative())) {
init_w();
init_r_B_O();
}
// here is what we need in the transition: an iterator that steps through
// the basic indices, where dereferencing
// yields an iterator through the corresponding row of D, restricted
// to the basic indices. This means that we select the principal minor of D
// corresponding to the current basis.
// To realize this, we transform B_O.begin() via the function h where
// h(i) = D_pairwise_iterator
// (B_O.begin(),
// D_pairwise_accessor(qp_D, i))
inv_M_B.transition
(boost::make_transform_iterator
(B_O.begin(),
boost::bind
(D_transition_creator_iterator(), B_O.begin(),
boost::bind (D_transition_creator_accessor(), qp_D, _1))));
}
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
transition( Tag_true)
{
inv_M_B.transition();
}
// ratio test
// ----------
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
ratio_test_init__2_D_Bj( Value_iterator, int, Tag_true)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // QP case
void QP_solver<Q, ET, Tags>::
ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j_, Tag_false)
{
if ( is_phaseII) {
ratio_test_init__2_D_Bj( two_D_Bj_it, j_,
Tag_false(), no_ineq);
}
}
template < typename Q, typename ET, typename Tags > inline // QP, no ineq.
void QP_solver<Q, ET, Tags>::
ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j_, Tag_false,
Tag_true )
{
// store exact version of `2 D_{B_O,j}'
D_pairwise_accessor d_accessor( qp_D, j_);
std::copy( D_pairwise_iterator( B_O.begin(), d_accessor),
D_pairwise_iterator( B_O.end (), d_accessor),
two_D_Bj_it);
}
template < typename Q, typename ET, typename Tags > inline // QP, has ineq
void QP_solver<Q, ET, Tags>::
ratio_test_init__2_D_Bj( Value_iterator two_D_Bj_it, int j_, Tag_false,
Tag_false)
{
// store exact version of `2 D_{B_O,j}'
if ( j_ < qp_n) { // original variable
ratio_test_init__2_D_Bj( two_D_Bj_it, j_, Tag_false(), Tag_true());
} else { // slack variable
std::fill_n( two_D_Bj_it, B_O.size(), et0);
}
}
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
ratio_test_1__q_x_O( Tag_true)
{
inv_M_B.multiply_x( A_Cj.begin(), q_x_O.begin());
}
template < typename Q, typename ET, typename Tags > inline // QP case
void QP_solver<Q, ET, Tags>::
ratio_test_1__q_x_O( Tag_false)
{
if ( is_phaseI) { // phase I
inv_M_B.multiply_x( A_Cj.begin(), q_x_O.begin());
} else { // phase II
inv_M_B.multiply ( A_Cj.begin(), two_D_Bj.begin(),
q_lambda.begin(), q_x_O.begin());
}
}
template < typename Q, typename ET, typename Tags > inline // no ineq.
void QP_solver<Q, ET, Tags>::
ratio_test_1__q_x_S( Tag_true)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // has ineq.
void QP_solver<Q, ET, Tags>::
ratio_test_1__q_x_S( Tag_false)
{
// A_S_BxB_O * q_x_O
multiply__A_S_BxB_O( q_x_O.begin(), q_x_S.begin());
// ( A_S_BxB_O * q_x_O) - A_S_Bxj
if ( j < qp_n) {
typedef typename std::iterator_traits<A_by_index_iterator>::value_type RT;
std::transform( q_x_S.begin(),
q_x_S.begin()+S_B.size(),
A_by_index_iterator( S_B.begin(),
A_by_index_accessor( *(qp_A + j))),
q_x_S.begin(),
boost::bind(std::minus<ET>(),
_1,
boost::bind(std::multiplies<ET>(), d,
boost::bind(
NT_converter<RT,ET>(),
_2))));
}
// q_x_S = -+ ( A_S_BxB_O * q_x_O - A_S_Bxj)
Value_iterator q_it = q_x_S.begin();
Index_iterator i_it;
for ( i_it = B_S.begin(); i_it != B_S.end(); ++i_it, ++q_it) {
if ( ! slack_A[ *i_it - qp_n].second) *q_it = -(*q_it);
}
}
template < typename Q, typename ET, typename Tags > inline // no check
void QP_solver<Q, ET, Tags>::
ratio_test_1__t_i( Index_iterator, Index_iterator,
Value_iterator, Value_iterator, Tag_true)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // check
void QP_solver<Q, ET, Tags>::
ratio_test_1__t_i( Index_iterator i_it, Index_iterator end_it,
Value_iterator x_it, Value_iterator q_it, Tag_false)
{
// check `t_i's
for ( ; i_it != end_it; ++i_it, ++x_it, ++q_it) {
// BLAND rule: In case the ratios are the same, only update if the new index
// is smaller. The special artificial variable is always made to leave first.
if ( (*q_it > et0) && (
(( *x_it * q_i) < ( x_i * *q_it)) ||
( (*i_it < i) && (i != art_s_i) && (( *x_it * q_i) == ( x_i * *q_it)) )
)
) {
i = *i_it; x_i = *x_it; q_i = *q_it;
}
}
}
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
ratio_test_1__t_j( Tag_true)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // QP case
void QP_solver<Q, ET, Tags>::
ratio_test_1__t_j( Tag_false)
{
if ( is_phaseII) {
// compute `nu' and `mu_j'
mu = mu_j(j);
nu = inv_M_B.inner_product( A_Cj.begin(), two_D_Bj.begin(),
q_lambda.begin(), q_x_O.begin());
if ( j < qp_n) { // original variable
nu -= d*ET( (*(qp_D + j))[ j]);
}
CGAL_qpe_assertion_msg(nu <= et0,
"nu <= et0 violated -- is your D matrix positive semidefinite?");
// check `t_j'
CGAL_qpe_assertion(mu != et0);
// bg: formula below compares abs values, assuming mu < 0
if ( ( nu < et0) && ( ( (mu < et0 ? mu : -mu) * q_i) > ( x_i * nu))) {
i = -1; q_i = et1;
}
}
}
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
ratio_test_2( Tag_true)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // no ineq.
void QP_solver<Q, ET, Tags>::
ratio_test_2__p( Tag_true)
{
// get column index of entering variable in basis
int col = in_B[ j];
CGAL_qpe_assertion( col >= 0);
col += l;
// get (last) column of `M_B^{-1}' (Note: `p_...' is stored in `q_...')
Value_iterator it;
int row;
unsigned int k;
for ( k = 0, row = 0, it = q_lambda.begin();
k < C.size();
++k, ++row, ++it ) {
*it = inv_M_B.entry( row, col);
}
for ( k = 0, row = l, it = q_x_O.begin();
k < B_O.size();
++k, ++row, ++it ) {
*it = inv_M_B.entry( row, col);
}
}
template < typename Q, typename ET, typename Tags > inline // has ineq.
void QP_solver<Q, ET, Tags>::
ratio_test_2__p( Tag_false)
{
Value_iterator v_it;
Index_iterator i_it;
// compute 'p_lambda' and 'p_x_O' (Note: `p_...' is stored in `q_...')
// -------------------------------------------------------------------
// type of entering variable
if ( j < qp_n) { // original
// use 'no_ineq' variant
ratio_test_2__p( Tag_true());
} else { // slack
j -= qp_n;
// get column A_{S_j,B_O}^T (i.e. row of A_{S_B,B_O})
int row = slack_A[ j].first;
bool sign = slack_A[ j].second;
for ( i_it = B_O.begin(), v_it = tmp_x.begin();
i_it != B_O.end();
++i_it, ++v_it ) {
*v_it = ( sign ?
*((*(qp_A+ *i_it))+ row) : - (*((*(qp_A + *i_it))+ row)));
}
// compute ( p_l | p_x_O )^T = M_B^{-1} * ( 0 | A_{S_j,B_O} )^T
std::fill_n( tmp_l.begin(), C.size(), et0);
inv_M_B.multiply( tmp_l .begin(), tmp_x .begin(),
q_lambda.begin(), q_x_O.begin());
j += qp_n;
}
// compute 'p_x_S'
// ---------------
// A_S_BxB_O * p_x_O
multiply__A_S_BxB_O( q_x_O.begin(), q_x_S.begin());
// p_x_S = +- ( A_S_BxB_O * p_x_O)
for ( i_it = B_S.begin(), v_it = q_x_S.begin();
i_it != B_S.end();
++i_it, ++v_it ) {
if ( ! slack_A[ *i_it - qp_n].second) *v_it = -(*v_it);
}
}
// update
// ------
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
update_1( Tag_true)
{
// replace leaving with entering variable
if ((i == j) && (i >= 0)) {
enter_and_leave_variable();
} else {
replace_variable();
}
}
template < typename Q, typename ET, typename Tags > inline // QP case
void QP_solver<Q, ET, Tags>::
update_1( Tag_false)
{
if ( is_phaseI) { // phase I
// replace leaving with entering variable
if ((i == j) && (i >= 0)) {
enter_and_leave_variable();
} else {
replace_variable();
}
} else { // phase II
if ((i == j) && (i >= 0)) {
enter_and_leave_variable();
} else {
if ( ( i >= 0) && basis_matrix_stays_regular()) {
// leave variable from basis, if
// - some leaving variable was found and
// - basis matrix stays regular
leave_variable();
} else {
// enter variable into basis, if
// - no leaving variable was found or
// - basis matrix would become singular when variable i leaves
if ( i < 0 ) {
enter_variable();
} else {
z_replace_variable();
}
}
}
}
}
template < typename Q, typename ET, typename Tags > inline // LP case
void QP_solver<Q, ET, Tags>::
update_2( Tag_true)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // no ineq.
void QP_solver<Q, ET, Tags>::
replace_variable( Tag_true)
{
replace_variable_original_original();
strategyP->leaving_basis( i);
}
template < typename Q, typename ET, typename Tags > inline // has ineq.
void QP_solver<Q, ET, Tags>::
replace_variable( Tag_false)
{
// determine type of variables
bool enter_original = ( (j < qp_n) || (j >= static_cast<int>( qp_n+slack_A.size())));
bool leave_original = ( (i < qp_n) || (i >= static_cast<int>( qp_n+slack_A.size())));
// update basis & basis inverse
if ( leave_original) {
if ( enter_original) { // orig <--> orig
replace_variable_original_original();
} else { // slack <--> orig
replace_variable_slack_original();
}
// special artificial variable removed?
if ( is_phaseI && ( i == art_s_i)) {
// remove the fake column - it corresponds
// to the special artificial variable which is
// (like all artificial variables) not needed
// anymore once it leaves the basis. Note:
// regular artificial variables are only removed
// from the problem after phase I
// art_s_i == -1 -> there is no special artificial variable
// art_s_i == -2 -> there was a special artificial variable,
// but has been removed
art_s_i = -2;
art_A.pop_back();
CGAL_qpe_assertion(in_B[in_B.size()-1] == -1); // really removed?
in_B.pop_back();
// BG: shouldn't the pricing strategy be notfied also here?
} else {
strategyP->leaving_basis( i);
}
} else {
if ( enter_original) { // orig <--> slack
replace_variable_original_slack();
} else { // slack <--> slack
replace_variable_slack_slack();
}
strategyP->leaving_basis( i);
}
}
template < typename Q, typename ET, typename Tags > inline
bool QP_solver<Q, ET, Tags>::
basis_matrix_stays_regular()
{
CGAL_qpe_assertion( is_phaseII);
int new_row, k;
if ( has_ineq && (i >= qp_n)) { // slack variable
new_row = slack_A[ i-qp_n].first;
A_row_by_index_accessor a_accessor =
boost::bind (A_accessor( qp_A, 0, qp_n), _1, new_row);
typedef typename std::iterator_traits<A_row_by_index_iterator>::value_type RT;
std::transform(A_row_by_index_iterator( B_O.begin(), a_accessor),
A_row_by_index_iterator( B_O.end (), a_accessor),
tmp_x.begin(),
NT_converter<RT,ET>());
inv_M_B.multiply( tmp_x.begin(), // dummy (not used)
tmp_x.begin(), tmp_l_2.begin(), tmp_x_2.begin(),
Tag_false(), // QP
Tag_false()); // ignore 1st argument
return ( -inv_M_B.inner_product_x( tmp_x_2.begin(), tmp_x.begin()) != et0);
} else { // check original variable
k = l+in_B[ i];
return ( inv_M_B.entry( k, k) != et0);
}
/* ToDo: check, if really not needed in 'update_1':
- basis has already minimal size or
|| ( B_O.size()==C.size())
*/
}
// current solution
// ----------------
template < typename Q, typename ET, typename Tags > inline // no inequalities, upper bounded
void QP_solver<Q, ET, Tags>::
compute__x_B_S( Tag_true /*has_equalities_only_and_full_rank*/,
Tag_false /*is_nonnegative*/)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // no inequalities, standard form
void QP_solver<Q, ET, Tags>::
compute__x_B_S( Tag_true /*has_equalities_only_and_full_rank*/,
Tag_true /*is_nonnegative*/)
{
// nop
}
template < typename Q, typename ET, typename Tags > inline // has inequalities, upper bounded
void QP_solver<Q, ET, Tags>::
compute__x_B_S( Tag_false /*has_equalities_only_and_full_rank*/,
Tag_false /*is_nonnegative*/)
{
// A_S_BxB_O * x_B_O
multiply__A_S_BxB_O( x_B_O.begin(), x_B_S.begin());
// b_S_B - ( A_S_BxB_O * x_B_O)
B_by_index_accessor b_accessor( qp_b);
std::transform( B_by_index_iterator( S_B.begin(), b_accessor),
B_by_index_iterator( S_B.end (), b_accessor),
x_B_S.begin(),
x_B_S.begin(),
compose2_2( std::minus<ET>(),
boost::bind1st( std::multiplies<ET>(), d),
Identity<ET>()));
// b_S_B - ( A_S_BxB_O * x_B_O) - r_S_B
std::transform(x_B_S.begin(), x_B_S.begin()+S_B.size(),
r_S_B.begin(), x_B_S.begin(),
compose2_2(std::minus<ET>(),
Identity<ET>(),
boost::bind1st( std::multiplies<ET>(), d)));
// x_B_S = +- ( b_S_B - A_S_BxB_O * x_B_O)
Value_iterator x_it = x_B_S.begin();
Index_iterator i_it;
for ( i_it = B_S.begin(); i_it != B_S.end(); ++i_it, ++x_it) {
if ( slack_A[ *i_it - qp_n].second) *x_it = -(*x_it);
}
}
namespace QP_solver_impl {
// Writing it with 5 boost::bind was becoming unreadable.
template<class ET, class RT>
struct submul {
ET const& d;
submul(ET const&d):d(d) {}
ET operator()(RT const&x, ET const&y) const {
NT_converter<RT, ET> cast;
return cast(x) * d - y;
}
};
}
template < typename Q, typename ET, typename Tags > inline // has inequalities, standard form
void QP_solver<Q, ET, Tags>::
compute__x_B_S( Tag_false /*has_equalities_only_and_full_rank*/,
Tag_true /*is_nonnegative*/)
{
// A_S_BxB_O * x_B_O
multiply__A_S_BxB_O( x_B_O.begin(), x_B_S.begin());
// b_S_B - ( A_S_BxB_O * x_B_O)
B_by_index_accessor b_accessor( qp_b);
typedef typename std::iterator_traits<B_by_index_iterator>::value_type RT;
std::transform( B_by_index_iterator( S_B.begin(), b_accessor),
B_by_index_iterator( S_B.end (), b_accessor),
x_B_S.begin(),
x_B_S.begin(),
QP_solver_impl::submul<ET,RT>(d));
// x_B_S = +- ( b_S_B - A_S_BxB_O * x_B_O)
Value_iterator x_it = x_B_S.begin();
Index_iterator i_it;
for ( i_it = B_S.begin(); i_it != B_S.end(); ++i_it, ++x_it) {
if ( slack_A[ *i_it - qp_n].second) *x_it = -(*x_it);
}
}
} //namespace CGAL
#include <CGAL/QP_solver/Unbounded_direction.h>
#include <CGAL/QP_solver/QP_solver_nonstandardform_impl.h>
#include <CGAL/QP_solver/QP_solver_bounds_impl.h>
#include <CGAL/QP_solver/QP_solver_impl.h>
#include <CGAL/enable_warnings.h>
#endif // CGAL_QP_SOLVER_H
// ===== EOF ==================================================================
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