/* -*- mode: C++; indent-tabs-mode: nil; -*-

  *

  * This file is a part of LEMON, a generic C++ optimization library.

  *

  * Copyright (C) 2003-2013

  * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

  * (Egervary Research Group on Combinatorial Optimization, EGRES).

  *

  * Permission to use, modify and distribute this software is granted

  * provided that this copyright notice appears in all copies. For

  * precise terms see the accompanying LICENSE file.

  *

  * This software is provided "AS IS" with no warranty of any kind,

  * express or implied, and with no claim as to its suitability for any

  * purpose.

  *

  */

 #ifndef LEMON_LP_BASE_H

 #define LEMON_LP_BASE_H

 #include<iostream>

 #include<vector>

 #include<map>

 #include<limits>

 #include<lemon/math.h>

 #include<lemon/error.h>

 #include<lemon/assert.h>

 #include<lemon/core.h>

 #include<lemon/bits/solver_bits.h>

 #include<lemon/bits/stl_iterators.h>

 ///\file

 ///\brief The interface of the LP solver interface.

 ///\ingroup lp_group

 namespace lemon {

   ///Common base class for LP and MIP solvers

   ///Usually this class is not used directly, please use one of the concrete

   ///implementations of the solver interface.

   ///\ingroup lp_group

   class LpBase {

   protected:

     _solver_bits::VarIndex _rows;

     _solver_bits::VarIndex _cols;

   public:

     ///Possible outcomes of an LP solving procedure

     enum SolveExitStatus {

       /// = 0. It means that the problem has been successfully solved: either

       ///an optimal solution has been found or infeasibility/unboundedness

       ///has been proved.

       SOLVED = 0,

       /// = 1. Any other case (including the case when some user specified

       ///limit has been exceeded).

       UNSOLVED = 1

     };

     ///Direction of the optimization

     enum Sense {

       /// Minimization

       MIN,

       /// Maximization

       MAX

     };

     ///Enum for \c messageLevel() parameter

     enum MessageLevel {

       /// No output (default value).

       MESSAGE_NOTHING,

       /// Error messages only.

       MESSAGE_ERROR,

       /// Warnings.

       MESSAGE_WARNING,

       /// Normal output.

       MESSAGE_NORMAL,

       /// Verbose output.

       MESSAGE_VERBOSE

     };

     ///The floating point type used by the solver

     typedef double Value;

     ///The infinity constant

     static const Value INF;

     ///The not a number constant

     static const Value NaN;

     friend class Col;

     friend class ColIt;

     friend class Row;

     friend class RowIt;

     ///Refer to a column of the LP.

     ///This type is used to refer to a column of the LP.

     ///

     ///Its value remains valid and correct even after the addition or erase of

     ///other columns.

     ///

     ///\note This class is similar to other Item types in LEMON, like

     ///Node and Arc types in digraph.

     class Col {

       friend class LpBase;

     protected:

       int _id;

       explicit Col(int id) : _id(id) {}

     public:

       typedef Value ExprValue;

       typedef True LpCol;

       /// Default constructor

       /// \warning The default constructor sets the Col to an

       /// undefined value.

       Col() {}

       /// Invalid constructor \& conversion.

       /// This constructor initializes the Col to be invalid.

       /// \sa Invalid for more details.

       Col(const Invalid&) : _id(-1) {}

       /// Equality operator

       /// Two \ref Col "Col"s are equal if and only if they point to

       /// the same LP column or both are invalid.

       bool operator==(Col c) const  {return _id == c._id;}

       /// Inequality operator

       /// \sa operator==(Col c)

       ///

       bool operator!=(Col c) const  {return _id != c._id;}

       /// Artificial ordering operator.

       /// To allow the use of this object in std::map or similar

       /// associative container we require this.

       ///

       /// \note This operator only have to define some strict ordering of

       /// the items; this order has nothing to do with the iteration

       /// ordering of the items.

       bool operator<(Col c) const  {return _id < c._id;}

     };

     ///Iterator for iterate over the columns of an LP problem

     /// Its usage is quite simple, for example, you can count the number

     /// of columns in an LP \c lp:

     ///\code

     /// int count=0;

     /// for (LpBase::ColIt c(lp); c!=INVALID; ++c) ++count;

     ///\endcode

     class ColIt : public Col {

       const LpBase *_solver;

     public:

       /// Default constructor

       /// \warning The default constructor sets the iterator

       /// to an undefined value.

       ColIt() {}

       /// Sets the iterator to the first Col

       /// Sets the iterator to the first Col.

       ///

       ColIt(const LpBase &solver) : _solver(&solver)

       {

         _solver->_cols.firstItem(_id);

       }

       /// Invalid constructor \& conversion

       /// Initialize the iterator to be invalid.

       /// \sa Invalid for more details.

       ColIt(const Invalid&) : Col(INVALID) {}

       /// Next column

       /// Assign the iterator to the next column.

       ///

       ColIt &operator++()

       {

         _solver->_cols.nextItem(_id);

         return *this;

       }

     };

     /// \brief Gets the collection of the columns of the LP problem.

     ///

     /// This function can be used for iterating on

     /// the columns of the LP problem. It returns a wrapped ColIt, which looks

     /// like an STL container (by having begin() and end())

     /// which you can use in range-based for loops, STL algorithms, etc.

     /// For example you can write:

     ///\code

     /// for(auto c: lp.cols())

     ///   doSomething(c);

     LemonRangeWrapper1<ColIt, LpBase> cols() {

       return LemonRangeWrapper1<ColIt, LpBase>(*this);

     }

     /// \brief Returns the ID of the column.

     static int id(const Col& col) { return col._id; }

     /// \brief Returns the column with the given ID.

     ///

     /// \pre The argument should be a valid column ID in the LP problem.

     static Col colFromId(int id) { return Col(id); }

     ///Refer to a row of the LP.

     ///This type is used to refer to a row of the LP.

     ///

     ///Its value remains valid and correct even after the addition or erase of

     ///other rows.

     ///

     ///\note This class is similar to other Item types in LEMON, like

     ///Node and Arc types in digraph.

     class Row {

       friend class LpBase;

     protected:

       int _id;

       explicit Row(int id) : _id(id) {}

     public:

       typedef Value ExprValue;

       typedef True LpRow;

       /// Default constructor

       /// \warning The default constructor sets the Row to an

       /// undefined value.

       Row() {}

       /// Invalid constructor \& conversion.

       /// This constructor initializes the Row to be invalid.

       /// \sa Invalid for more details.

       Row(const Invalid&) : _id(-1) {}

       /// Equality operator

       /// Two \ref Row "Row"s are equal if and only if they point to

       /// the same LP row or both are invalid.

       bool operator==(Row r) const  {return _id == r._id;}

       /// Inequality operator

       /// \sa operator==(Row r)

       ///

       bool operator!=(Row r) const  {return _id != r._id;}

       /// Artificial ordering operator.

       /// To allow the use of this object in std::map or similar

       /// associative container we require this.

       ///

       /// \note This operator only have to define some strict ordering of

       /// the items; this order has nothing to do with the iteration

       /// ordering of the items.

       bool operator<(Row r) const  {return _id < r._id;}

     };

     ///Iterator for iterate over the rows of an LP problem

     /// Its usage is quite simple, for example, you can count the number

     /// of rows in an LP \c lp:

     ///\code

     /// int count=0;

     /// for (LpBase::RowIt c(lp); c!=INVALID; ++c) ++count;

     ///\endcode

     class RowIt : public Row {

       const LpBase *_solver;

     public:

       /// Default constructor

       /// \warning The default constructor sets the iterator

       /// to an undefined value.

       RowIt() {}

       /// Sets the iterator to the first Row

       /// Sets the iterator to the first Row.

       ///

       RowIt(const LpBase &solver) : _solver(&solver)

       {

         _solver->_rows.firstItem(_id);

       }

       /// Invalid constructor \& conversion

       /// Initialize the iterator to be invalid.

       /// \sa Invalid for more details.

       RowIt(const Invalid&) : Row(INVALID) {}

       /// Next row

       /// Assign the iterator to the next row.

       ///

       RowIt &operator++()

       {

         _solver->_rows.nextItem(_id);

         return *this;

       }

     };

     /// \brief Gets the collection of the rows of the LP problem.

     ///

     /// This function can be used for iterating on

     /// the rows of the LP problem. It returns a wrapped RowIt, which looks

     /// like an STL container (by having begin() and end())

     /// which you can use in range-based for loops, STL algorithms, etc.

     /// For example you can write:

     ///\code

     /// for(auto c: lp.rows())

     ///   doSomething(c);

     LemonRangeWrapper1<RowIt, LpBase> rows() {

       return LemonRangeWrapper1<RowIt, LpBase>(*this);

     }

     /// \brief Returns the ID of the row.

     static int id(const Row& row) { return row._id; }

     /// \brief Returns the row with the given ID.

     ///

     /// \pre The argument should be a valid row ID in the LP problem.

     static Row rowFromId(int id) { return Row(id); }

   public:

     ///Linear expression of variables and a constant component

     ///This data structure stores a linear expression of the variables

     ///(\ref Col "Col"s) and also has a constant component.

     ///

     ///There are several ways to access and modify the contents of this

     ///container.

     ///\code

     ///e[v]=5;

     ///e[v]+=12;

     ///e.erase(v);

     ///\endcode

     ///or you can also iterate through its elements.

     ///\code

     ///double s=0;

     ///for(LpBase::Expr::ConstCoeffIt i(e);i!=INVALID;++i)

     ///  s+=*i * primal(i);

     ///\endcode

     ///(This code computes the primal value of the expression).

     ///- Numbers (<tt>double</tt>'s)

     ///and variables (\ref Col "Col"s) directly convert to an

     ///\ref Expr and the usual linear operations are defined, so

     ///\code

     ///v+w

     ///2*v-3.12*(v-w/2)+2

     ///v*2.1+(3*v+(v*12+w+6)*3)/2

     ///\endcode

     ///are valid expressions.

     ///The usual assignment operations are also defined.

     ///\code

     ///e=v+w;

     ///e+=2*v-3.12*(v-w/2)+2;

     ///e*=3.4;

     ///e/=5;

     ///\endcode

     ///- The constant member can be set and read by dereference

     ///  operator (unary *)

     ///

     ///\code

     ///*e=12;

     ///double c=*e;

     ///\endcode

     ///

     ///\sa Constr

     class Expr {

       friend class LpBase;

     public:

       /// The key type of the expression

       typedef LpBase::Col Key;

       /// The value type of the expression

       typedef LpBase::Value Value;

     protected:

       Value const_comp;

       std::map<int, Value> comps;

     public:

       typedef True SolverExpr;

       /// Default constructor

       /// Construct an empty expression, the coefficients and

       /// the constant component are initialized to zero.

       Expr() : const_comp(0) {}

       /// Construct an expression from a column

       /// Construct an expression, which has a term with \c c variable

       /// and 1.0 coefficient.

       Expr(const Col &c) : const_comp(0) {

         typedef std::map<int, Value>::value_type pair_type;

         comps.insert(pair_type(id(c), 1));

       }

       /// Construct an expression from a constant

       /// Construct an expression, which's constant component is \c v.

       ///

       Expr(const Value &v) : const_comp(v) {}

       /// Returns the coefficient of the column

       Value operator[](const Col& c) const {

         std::map<int, Value>::const_iterator it=comps.find(id(c));

         if (it != comps.end()) {

           return it->second;

         } else {

           return 0;

         }

       }

       /// Returns the coefficient of the column

       Value& operator[](const Col& c) {

         return comps[id(c)];

       }

       /// Sets the coefficient of the column

       void set(const Col &c, const Value &v) {

         if (v != 0.0) {

           typedef std::map<int, Value>::value_type pair_type;

           comps.insert(pair_type(id(c), v));

         } else {

           comps.erase(id(c));

         }

       }

       /// Returns the constant component of the expression

       Value& operator*() { return const_comp; }

       /// Returns the constant component of the expression

       const Value& operator*() const { return const_comp; }

       /// \brief Removes the coefficients which's absolute value does

       /// not exceed \c epsilon. It also sets to zero the constant

       /// component, if it does not exceed epsilon in absolute value.

       void simplify(Value epsilon = 0.0) {

         std::map<int, Value>::iterator it=comps.begin();

         while (it != comps.end()) {

           std::map<int, Value>::iterator jt=it;

           ++jt;

           if (std::fabs((*it).second) <= epsilon) comps.erase(it);

           it=jt;

         }

         if (std::fabs(const_comp) <= epsilon) const_comp = 0;

       }

       void simplify(Value epsilon = 0.0) const {

         const_cast<Expr*>(this)->simplify(epsilon);

       }

       ///Sets all coefficients and the constant component to 0.

       void clear() {

         comps.clear();

         const_comp=0;

       }

       ///Compound assignment

       Expr &operator+=(const Expr &e) {

         for (std::map<int, Value>::const_iterator it=e.comps.begin();

              it!=e.comps.end(); ++it)

           comps[it->first]+=it->second;

         const_comp+=e.const_comp;

         return *this;

       }

       ///Compound assignment

       Expr &operator-=(const Expr &e) {

         for (std::map<int, Value>::const_iterator it=e.comps.begin();

              it!=e.comps.end(); ++it)

           comps[it->first]-=it->second;

         const_comp-=e.const_comp;

         return *this;

       }

       ///Multiply with a constant

       Expr &operator*=(const Value &v) {

         for (std::map<int, Value>::iterator it=comps.begin();

              it!=comps.end(); ++it)

           it->second*=v;

         const_comp*=v;

         return *this;

       }

       ///Division with a constant

       Expr &operator/=(const Value &c) {

         for (std::map<int, Value>::iterator it=comps.begin();

              it!=comps.end(); ++it)

           it->second/=c;

         const_comp/=c;

         return *this;

       }

       ///Iterator over the expression

       ///The iterator iterates over the terms of the expression.

       ///

       ///\code

       ///double s=0;

       ///for(LpBase::Expr::CoeffIt i(e);i!=INVALID;++i)

       ///  s+= *i * primal(i);

       ///\endcode

       class CoeffIt {

       private:

         std::map<int, Value>::iterator _it, _end;

       public:

         /// Sets the iterator to the first term

         /// Sets the iterator to the first term of the expression.

         ///

         CoeffIt(Expr& e)

           : _it(e.comps.begin()), _end(e.comps.end()){}

         /// Convert the iterator to the column of the term

         operator Col() const {

           return colFromId(_it->first);

         }

         /// Returns the coefficient of the term

         Value& operator*() { return _it->second; }

         /// Returns the coefficient of the term

         const Value& operator*() const { return _it->second; }

         /// Next term

         /// Assign the iterator to the next term.

         ///

         CoeffIt& operator++() { ++_it; return *this; }

         /// Equality operator

         bool operator==(Invalid) const { return _it == _end; }

         /// Inequality operator

         bool operator!=(Invalid) const { return _it != _end; }

       };

       /// Const iterator over the expression

       ///The iterator iterates over the terms of the expression.

       ///

       ///\code

       ///double s=0;

       ///for(LpBase::Expr::ConstCoeffIt i(e);i!=INVALID;++i)

       ///  s+=*i * primal(i);

       ///\endcode

       class ConstCoeffIt {

       private:

         std::map<int, Value>::const_iterator _it, _end;

       public:

         /// Sets the iterator to the first term

         /// Sets the iterator to the first term of the expression.

         ///

         ConstCoeffIt(const Expr& e)

           : _it(e.comps.begin()), _end(e.comps.end()){}

         /// Convert the iterator to the column of the term

         operator Col() const {

           return colFromId(_it->first);

         }

         /// Returns the coefficient of the term

         const Value& operator*() const { return _it->second; }

         /// Next term

         /// Assign the iterator to the next term.

         ///

         ConstCoeffIt& operator++() { ++_it; return *this; }

         /// Equality operator

         bool operator==(Invalid) const { return _it == _end; }

         /// Inequality operator

         bool operator!=(Invalid) const { return _it != _end; }

       };

     };

     ///Linear constraint

     ///This data stucture represents a linear constraint in the LP.

     ///Basically it is a linear expression with a lower or an upper bound

     ///(or both). These parts of the constraint can be obtained by the member

     ///functions \ref expr(), \ref lowerBound() and \ref upperBound(),

     ///respectively.

     ///There are two ways to construct a constraint.

     ///- You can set the linear expression and the bounds directly

     ///  by the functions above.

     ///- The operators <tt>\<=</tt>, <tt>==</tt> and  <tt>\>=</tt>

     ///  are defined between expressions, or even between constraints whenever

     ///  it makes sense. Therefore if \c e and \c f are linear expressions and

     ///  \c s and \c t are numbers, then the followings are valid expressions

     ///  and thus they can be used directly e.g. in \ref addRow() whenever

     ///  it makes sense.

     ///\code

     ///  e<=s

     ///  e<=f

     ///  e==f

     ///  s<=e<=t

     ///  e>=t

     ///\endcode

     ///\warning The validity of a constraint is checked only at run

     ///time, so e.g. \ref addRow(<tt>x[1]\<=x[2]<=5</tt>) will

     ///compile, but will fail an assertion.

     class Constr

     {

     public:

       typedef LpBase::Expr Expr;

       typedef Expr::Key Key;

       typedef Expr::Value Value;

     protected:

       Expr _expr;

       Value _lb,_ub;

     public:

       ///\e

       Constr() : _expr(), _lb(NaN), _ub(NaN) {}

       ///\e

       Constr(Value lb, const Expr &e, Value ub) :

         _expr(e), _lb(lb), _ub(ub) {}

       Constr(const Expr &e) :

         _expr(e), _lb(NaN), _ub(NaN) {}

       ///\e

       void clear()

       {

         _expr.clear();

         _lb=_ub=NaN;

       }

       ///Reference to the linear expression

       Expr &expr() { return _expr; }

       ///Cont reference to the linear expression

       const Expr &expr() const { return _expr; }

       ///Reference to the lower bound.

       ///\return

       ///- \ref INF "INF": the constraint is lower unbounded.

       ///- \ref NaN "NaN": lower bound has not been set.

       ///- finite number: the lower bound

       Value &lowerBound() { return _lb; }

       ///The const version of \ref lowerBound()

       const Value &lowerBound() const { return _lb; }

       ///Reference to the upper bound.

       ///\return

       ///- \ref INF "INF": the constraint is upper unbounded.

       ///- \ref NaN "NaN": upper bound has not been set.

       ///- finite number: the upper bound

       Value &upperBound() { return _ub; }

       ///The const version of \ref upperBound()

       const Value &upperBound() const { return _ub; }

       ///Is the constraint lower bounded?

       bool lowerBounded() const {

         return _lb != -INF && !isNaN(_lb);

       }

       ///Is the constraint upper bounded?

       bool upperBounded() const {

         return _ub != INF && !isNaN(_ub);

       }

     };

     ///Linear expression of rows

     ///This data structure represents a column of the matrix,

     ///thas is it strores a linear expression of the dual variables

     ///(\ref Row "Row"s).

     ///

     ///There are several ways to access and modify the contents of this

     ///container.

     ///\code

     ///e[v]=5;

     ///e[v]+=12;

     ///e.erase(v);

     ///\endcode

     ///or you can also iterate through its elements.

     ///\code

     ///double s=0;

     ///for(LpBase::DualExpr::ConstCoeffIt i(e);i!=INVALID;++i)

     ///  s+=*i;

     ///\endcode

     ///(This code computes the sum of all coefficients).

     ///- Numbers (<tt>double</tt>'s)

     ///and variables (\ref Row "Row"s) directly convert to an

     ///\ref DualExpr and the usual linear operations are defined, so

     ///\code

     ///v+w

     ///2*v-3.12*(v-w/2)

     ///v*2.1+(3*v+(v*12+w)*3)/2

     ///\endcode

     ///are valid \ref DualExpr dual expressions.

     ///The usual assignment operations are also defined.

     ///\code

     ///e=v+w;

     ///e+=2*v-3.12*(v-w/2);

     ///e*=3.4;

     ///e/=5;

     ///\endcode

     ///

     ///\sa Expr

     class DualExpr {

       friend class LpBase;

     public:

       /// The key type of the expression

       typedef LpBase::Row Key;

       /// The value type of the expression

       typedef LpBase::Value Value;

     protected:

       std::map<int, Value> comps;

     public:

       typedef True SolverExpr;

       /// Default constructor

       /// Construct an empty expression, the coefficients are

       /// initialized to zero.

       DualExpr() {}

       /// Construct an expression from a row

       /// Construct an expression, which has a term with \c r dual

       /// variable and 1.0 coefficient.

       DualExpr(const Row &r) {

         typedef std::map<int, Value>::value_type pair_type;

         comps.insert(pair_type(id(r), 1));

       }

       /// Returns the coefficient of the row

       Value operator[](const Row& r) const {

         std::map<int, Value>::const_iterator it = comps.find(id(r));

         if (it != comps.end()) {

           return it->second;

         } else {

           return 0;

         }

       }

       /// Returns the coefficient of the row

       Value& operator[](const Row& r) {

         return comps[id(r)];

       }

       /// Sets the coefficient of the row

       void set(const Row &r, const Value &v) {

         if (v != 0.0) {

           typedef std::map<int, Value>::value_type pair_type;

           comps.insert(pair_type(id(r), v));

         } else {

           comps.erase(id(r));

         }

       }

       /// \brief Removes the coefficients which's absolute value does

       /// not exceed \c epsilon.

       void simplify(Value epsilon = 0.0) {

         std::map<int, Value>::iterator it=comps.begin();

         while (it != comps.end()) {

           std::map<int, Value>::iterator jt=it;

           ++jt;

           if (std::fabs((*it).second) <= epsilon) comps.erase(it);

           it=jt;

         }

       }

       void simplify(Value epsilon = 0.0) const {

         const_cast<DualExpr*>(this)->simplify(epsilon);

       }

       ///Sets all coefficients to 0.

       void clear() {

         comps.clear();

       }

       ///Compound assignment

       DualExpr &operator+=(const DualExpr &e) {

         for (std::map<int, Value>::const_iterator it=e.comps.begin();

              it!=e.comps.end(); ++it)

           comps[it->first]+=it->second;

         return *this;

       }

       ///Compound assignment

       DualExpr &operator-=(const DualExpr &e) {

         for (std::map<int, Value>::const_iterator it=e.comps.begin();

              it!=e.comps.end(); ++it)

           comps[it->first]-=it->second;

         return *this;

       }

       ///Multiply with a constant

       DualExpr &operator*=(const Value &v) {

         for (std::map<int, Value>::iterator it=comps.begin();

              it!=comps.end(); ++it)

           it->second*=v;

         return *this;

       }

       ///Division with a constant

       DualExpr &operator/=(const Value &v) {

         for (std::map<int, Value>::iterator it=comps.begin();

              it!=comps.end(); ++it)

           it->second/=v;

         return *this;

       }

       ///Iterator over the expression

       ///The iterator iterates over the terms of the expression.

       ///

       ///\code

       ///double s=0;

       ///for(LpBase::DualExpr::CoeffIt i(e);i!=INVALID;++i)

       ///  s+= *i * dual(i);

       ///\endcode

       class CoeffIt {

       private:

         std::map<int, Value>::iterator _it, _end;

       public:

         /// Sets the iterator to the first term

         /// Sets the iterator to the first term of the expression.

         ///

         CoeffIt(DualExpr& e)

           : _it(e.comps.begin()), _end(e.comps.end()){}

         /// Convert the iterator to the row of the term

         operator Row() const {

           return rowFromId(_it->first);

         }

         /// Returns the coefficient of the term

         Value& operator*() { return _it->second; }

         /// Returns the coefficient of the term

         const Value& operator*() const { return _it->second; }

         /// Next term

         /// Assign the iterator to the next term.

         ///

         CoeffIt& operator++() { ++_it; return *this; }

         /// Equality operator

         bool operator==(Invalid) const { return _it == _end; }

         /// Inequality operator

         bool operator!=(Invalid) const { return _it != _end; }

       };

       ///Iterator over the expression

       ///The iterator iterates over the terms of the expression.

       ///

       ///\code

       ///double s=0;

       ///for(LpBase::DualExpr::ConstCoeffIt i(e);i!=INVALID;++i)

       ///  s+= *i * dual(i);

       ///\endcode

       class ConstCoeffIt {

       private:

         std::map<int, Value>::const_iterator _it, _end;

       public:

         /// Sets the iterator to the first term

         /// Sets the iterator to the first term of the expression.

         ///

         ConstCoeffIt(const DualExpr& e)

           : _it(e.comps.begin()), _end(e.comps.end()){}

         /// Convert the iterator to the row of the term

         operator Row() const {

           return rowFromId(_it->first);

         }

         /// Returns the coefficient of the term

         const Value& operator*() const { return _it->second; }

         /// Next term

         /// Assign the iterator to the next term.

         ///

         ConstCoeffIt& operator++() { ++_it; return *this; }

         /// Equality operator

         bool operator==(Invalid) const { return _it == _end; }

         /// Inequality operator

         bool operator!=(Invalid) const { return _it != _end; }

       };

     };

   protected:

     class InsertIterator {

     private:

       std::map<int, Value>& _host;

       const _solver_bits::VarIndex& _index;

     public:

       typedef std::output_iterator_tag iterator_category;

       typedef void difference_type;

       typedef void value_type;

       typedef void reference;

       typedef void pointer;

       InsertIterator(std::map<int, Value>& host,

                    const _solver_bits::VarIndex& index)

         : _host(host), _index(index) {}

       InsertIterator& operator=(const std::pair<int, Value>& value) {

         typedef std::map<int, Value>::value_type pair_type;

         _host.insert(pair_type(_index[value.first], value.second));

         return *this;

       }

       InsertIterator& operator*() { return *this; }

       InsertIterator& operator++() { return *this; }

       InsertIterator operator++(int) { return *this; }

     };

     class ExprIterator {

     private:

       std::map<int, Value>::const_iterator _host_it;

       const _solver_bits::VarIndex& _index;

     public:

       typedef std::bidirectional_iterator_tag iterator_category;

       typedef std::ptrdiff_t difference_type;

       typedef const std::pair<int, Value> value_type;

       typedef value_type reference;

       class pointer {

       public:

         pointer(value_type& _value) : value(_value) {}

         value_type* operator->() { return &value; }

       private:

         value_type value;

       };

       ExprIterator(const std::map<int, Value>::const_iterator& host_it,

                    const _solver_bits::VarIndex& index)

         : _host_it(host_it), _index(index) {}

       reference operator*() {

         return std::make_pair(_index(_host_it->first), _host_it->second);

       }

       pointer operator->() {

         return pointer(operator*());

       }

       ExprIterator& operator++() { ++_host_it; return *this; }

       ExprIterator operator++(int) {

         ExprIterator tmp(*this); ++_host_it; return tmp;

       }

       ExprIterator& operator--() { --_host_it; return *this; }

       ExprIterator operator--(int) {

         ExprIterator tmp(*this); --_host_it; return tmp;

       }

       bool operator==(const ExprIterator& it) const {

         return _host_it == it._host_it;

       }

       bool operator!=(const ExprIterator& it) const {

         return _host_it != it._host_it;

       }

     };

   protected:

     //Abstract virtual functions

     virtual int _addColId(int col) { return _cols.addIndex(col); }

     virtual int _addRowId(int row) { return _rows.addIndex(row); }

     virtual void _eraseColId(int col) { _cols.eraseIndex(col); }

     virtual void _eraseRowId(int row) { _rows.eraseIndex(row); }

     virtual int _addCol() = 0;

     virtual int _addRow() = 0;

     virtual int _addRow(Value l, ExprIterator b, ExprIterator e, Value u) {

       int row = _addRow();

       _setRowCoeffs(row, b, e);

       _setRowLowerBound(row, l);

       _setRowUpperBound(row, u);

       return row;

     }

     virtual void _eraseCol(int col) = 0;

     virtual void _eraseRow(int row) = 0;

     virtual void _getColName(int col, std::string& name) const = 0;

     virtual void _setColName(int col, const std::string& name) = 0;

     virtual int _colByName(const std::string& name) const = 0;

     virtual void _getRowName(int row, std::string& name) const = 0;

     virtual void _setRowName(int row, const std::string& name) = 0;

     virtual int _rowByName(const std::string& name) const = 0;

     virtual void _setRowCoeffs(int i, ExprIterator b, ExprIterator e) = 0;

     virtual void _getRowCoeffs(int i, InsertIterator b) const = 0;

     virtual void _setColCoeffs(int i, ExprIterator b, ExprIterator e) = 0;

     virtual void _getColCoeffs(int i, InsertIterator b) const = 0;

     virtual void _setCoeff(int row, int col, Value value) = 0;

     virtual Value _getCoeff(int row, int col) const = 0;

     virtual void _setColLowerBound(int i, Value value) = 0;

     virtual Value _getColLowerBound(int i) const = 0;

     virtual void _setColUpperBound(int i, Value value) = 0;

     virtual Value _getColUpperBound(int i) const = 0;

     virtual void _setRowLowerBound(int i, Value value) = 0;

     virtual Value _getRowLowerBound(int i) const = 0;

     virtual void _setRowUpperBound(int i, Value value) = 0;

     virtual Value _getRowUpperBound(int i) const = 0;

     virtual void _setObjCoeffs(ExprIterator b, ExprIterator e) = 0;

     virtual void _getObjCoeffs(InsertIterator b) const = 0;

     virtual void _setObjCoeff(int i, Value obj_coef) = 0;

     virtual Value _getObjCoeff(int i) const = 0;

     virtual void _setSense(Sense) = 0;

     virtual Sense _getSense() const = 0;

     virtual void _clear() = 0;

     virtual const char* _solverName() const = 0;

     virtual void _messageLevel(MessageLevel level) = 0;

     //Own protected stuff

     //Constant component of the objective function

     Value obj_const_comp;

     LpBase() : _rows(), _cols(), obj_const_comp(0) {}

   public:

     ///Unsupported file format exception

     class UnsupportedFormatError : public Exception

     {

       std::string _format;

       mutable std::string _what;

     public:

       explicit UnsupportedFormatError(std::string format) throw()

         : _format(format) { }

       virtual ~UnsupportedFormatError() throw() {}

       virtual const char* what() const throw() {

         try {

           _what.clear();

           std::ostringstream oss;

           oss << "lemon::UnsupportedFormatError: " << _format;

           _what = oss.str();

         }

         catch (...) {}

         if (!_what.empty()) return _what.c_str();

         else return "lemon::UnsupportedFormatError";

       }

     };

   protected:

     virtual void _write(std::string, std::string format) const

     {

       throw UnsupportedFormatError(format);

     }

   public:

     /// Virtual destructor

     virtual ~LpBase() {}

     ///Gives back the name of the solver.

     const char* solverName() const {return _solverName();}

     ///\name Build Up and Modify the LP

     ///@{

     ///Add a new empty column (i.e a new variable) to the LP

     Col addCol() { Col c; c._id = _addColId(_addCol()); return c;}

     ///\brief Adds several new columns (i.e variables) at once

     ///

     ///This magic function takes a container as its argument and fills

     ///its elements with new columns (i.e. variables)

     ///\param t can be

     ///- a standard STL compatible iterable container with

     ///\ref Col as its \c values_type like

     ///\code

     ///std::vector<LpBase::Col>

     ///std::list<LpBase::Col>

     ///\endcode

     ///- a standard STL compatible iterable container with

     ///\ref Col as its \c mapped_type like

     ///\code

     ///std::map<AnyType,LpBase::Col>

     ///\endcode

     ///- an iterable lemon \ref concepts::WriteMap "write map" like

     ///\code

     ///ListGraph::NodeMap<LpBase::Col>

     ///ListGraph::ArcMap<LpBase::Col>

     ///\endcode

     ///\return The number of the created column.

 #ifdef DOXYGEN

     template<class T>

     int addColSet(T &t) { return 0;}

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpCol,int>::type

     addColSet(T &t,dummy<0> = 0) {

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {*i=addCol();s++;}

       return s;

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpCol,

                        int>::type

     addColSet(T &t,dummy<1> = 1) {

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

         i->second=addCol();

         s++;

       }

       return s;

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpCol,

                        int>::type

     addColSet(T &t,dummy<2> = 2) {

       int s=0;

       for(typename T::MapIt i(t); i!=INVALID; ++i)

         {

           i.set(addCol());

           s++;

         }

       return s;

     }

 #endif

     ///Set a column (i.e a dual constraint) of the LP

     ///\param c is the column to be modified

     ///\param e is a dual linear expression (see \ref DualExpr)

     ///a better one.

     void col(Col c, const DualExpr &e) {

       e.simplify();

       _setColCoeffs(_cols(id(c)), ExprIterator(e.comps.begin(), _rows),

                     ExprIterator(e.comps.end(), _rows));

     }

     ///Get a column (i.e a dual constraint) of the LP

     ///\param c is the column to get

     ///\return the dual expression associated to the column

     DualExpr col(Col c) const {

       DualExpr e;

       _getColCoeffs(_cols(id(c)), InsertIterator(e.comps, _rows));

       return e;

     }

     ///Add a new column to the LP

     ///\param e is a dual linear expression (see \ref DualExpr)

     ///\param o is the corresponding component of the objective

     ///function. It is 0 by default.

     ///\return The created column.

     Col addCol(const DualExpr &e, Value o = 0) {

       Col c=addCol();

       col(c,e);

       objCoeff(c,o);

       return c;

     }

     ///Add a new empty row (i.e a new constraint) to the LP

     ///This function adds a new empty row (i.e a new constraint) to the LP.

     ///\return The created row

     Row addRow() { Row r; r._id = _addRowId(_addRow()); return r;}

     ///\brief Add several new rows (i.e constraints) at once

     ///

     ///This magic function takes a container as its argument and fills

     ///its elements with new row (i.e. variables)

     ///\param t can be

     ///- a standard STL compatible iterable container with

     ///\ref Row as its \c values_type like

     ///\code

     ///std::vector<LpBase::Row>

     ///std::list<LpBase::Row>

     ///\endcode

     ///- a standard STL compatible iterable container with

     ///\ref Row as its \c mapped_type like

     ///\code

     ///std::map<AnyType,LpBase::Row>

     ///\endcode

     ///- an iterable lemon \ref concepts::WriteMap "write map" like

     ///\code

     ///ListGraph::NodeMap<LpBase::Row>

     ///ListGraph::ArcMap<LpBase::Row>

     ///\endcode

     ///\return The number of rows created.

 #ifdef DOXYGEN

     template<class T>

     int addRowSet(T &t) { return 0;}

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpRow,int>::type

     addRowSet(T &t, dummy<0> = 0) {

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {*i=addRow();s++;}

       return s;

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpRow, int>::type

     addRowSet(T &t, dummy<1> = 1) {

       int s=0;

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

         i->second=addRow();

         s++;

       }

       return s;

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpRow, int>::type

     addRowSet(T &t, dummy<2> = 2) {

       int s=0;

       for(typename T::MapIt i(t); i!=INVALID; ++i)

         {

           i.set(addRow());

           s++;

         }

       return s;

     }

 #endif

     ///Set a row (i.e a constraint) of the LP

     ///\param r is the row to be modified

     ///\param l is lower bound (-\ref INF means no bound)

     ///\param e is a linear expression (see \ref Expr)

     ///\param u is the upper bound (\ref INF means no bound)

     void row(Row r, Value l, const Expr &e, Value u) {

       e.simplify();

       _setRowCoeffs(_rows(id(r)), ExprIterator(e.comps.begin(), _cols),

                     ExprIterator(e.comps.end(), _cols));

       _setRowLowerBound(_rows(id(r)),l - *e);

       _setRowUpperBound(_rows(id(r)),u - *e);

     }

     ///Set a row (i.e a constraint) of the LP

     ///\param r is the row to be modified

     ///\param c is a linear expression (see \ref Constr)

     void row(Row r, const Constr &c) {

       row(r, c.lowerBounded()?c.lowerBound():-INF,

           c.expr(), c.upperBounded()?c.upperBound():INF);

     }

     ///Get a row (i.e a constraint) of the LP

     ///\param r is the row to get

     ///\return the expression associated to the row

     Expr row(Row r) const {

       Expr e;

       _getRowCoeffs(_rows(id(r)), InsertIterator(e.comps, _cols));

       return e;

     }

     ///Add a new row (i.e a new constraint) to the LP

     ///\param l is the lower bound (-\ref INF means no bound)

     ///\param e is a linear expression (see \ref Expr)

     ///\param u is the upper bound (\ref INF means no bound)

     ///\return The created row.

     Row addRow(Value l,const Expr &e, Value u) {

       Row r;

       e.simplify();

       r._id = _addRowId(_addRow(l - *e, ExprIterator(e.comps.begin(), _cols),

                                 ExprIterator(e.comps.end(), _cols), u - *e));

       return r;

     }

     ///Add a new row (i.e a new constraint) to the LP

     ///\param c is a linear expression (see \ref Constr)

     ///\return The created row.

     Row addRow(const Constr &c) {

       Row r;

       c.expr().simplify();

       r._id = _addRowId(_addRow(c.lowerBounded()?c.lowerBound()-*c.expr():-INF,

                                 ExprIterator(c.expr().comps.begin(), _cols),

                                 ExprIterator(c.expr().comps.end(), _cols),

                                 c.upperBounded()?c.upperBound()-*c.expr():INF));

       return r;

     }

     ///Erase a column (i.e a variable) from the LP

     ///\param c is the column to be deleted

     void erase(Col c) {

       _eraseCol(_cols(id(c)));

       _eraseColId(_cols(id(c)));

     }

     ///Erase a row (i.e a constraint) from the LP

     ///\param r is the row to be deleted

     void erase(Row r) {

       _eraseRow(_rows(id(r)));

       _eraseRowId(_rows(id(r)));

     }

     /// Get the name of a column

     ///\param c is the coresponding column

     ///\return The name of the colunm

     std::string colName(Col c) const {

       std::string name;

       _getColName(_cols(id(c)), name);

       return name;

     }

     /// Set the name of a column

     ///\param c is the coresponding column

     ///\param name The name to be given

     void colName(Col c, const std::string& name) {

       _setColName(_cols(id(c)), name);

     }

     /// Get the column by its name

     ///\param name The name of the column

     ///\return the proper column or \c INVALID

     Col colByName(const std::string& name) const {

       int k = _colByName(name);

       return k != -1 ? Col(_cols[k]) : Col(INVALID);

     }

     /// Get the name of a row

     ///\param r is the coresponding row

     ///\return The name of the row

     std::string rowName(Row r) const {

       std::string name;

       _getRowName(_rows(id(r)), name);

       return name;

     }

     /// Set the name of a row

     ///\param r is the coresponding row

     ///\param name The name to be given

     void rowName(Row r, const std::string& name) {

       _setRowName(_rows(id(r)), name);

     }

     /// Get the row by its name

     ///\param name The name of the row

     ///\return the proper row or \c INVALID

     Row rowByName(const std::string& name) const {

       int k = _rowByName(name);

       return k != -1 ? Row(_rows[k]) : Row(INVALID);

     }

     /// Set an element of the coefficient matrix of the LP

     ///\param r is the row of the element to be modified

     ///\param c is the column of the element to be modified

     ///\param val is the new value of the coefficient

     void coeff(Row r, Col c, Value val) {

       _setCoeff(_rows(id(r)),_cols(id(c)), val);

     }

     /// Get an element of the coefficient matrix of the LP

     ///\param r is the row of the element

     ///\param c is the column of the element

     ///\return the corresponding coefficient

     Value coeff(Row r, Col c) const {

       return _getCoeff(_rows(id(r)),_cols(id(c)));

     }

     /// Set the lower bound of a column (i.e a variable)

     /// The lower bound of a variable (column) has to be given by an

     /// extended number of type Value, i.e. a finite number of type

     /// Value or -\ref INF.

     void colLowerBound(Col c, Value value) {

       _setColLowerBound(_cols(id(c)),value);

     }

     /// Get the lower bound of a column (i.e a variable)

     /// This function returns the lower bound for column (variable) \c c

     /// (this might be -\ref INF as well).

     ///\return The lower bound for column \c c

     Value colLowerBound(Col c) const {

       return _getColLowerBound(_cols(id(c)));

     }

     ///\brief Set the lower bound of  several columns

     ///(i.e variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and applies the function on all of its elements.

     ///The lower bound of a variable (column) has to be given by an

     ///extended number of type Value, i.e. a finite number of type

     ///Value or -\ref INF.

 #ifdef DOXYGEN

     template<class T>

     void colLowerBound(T &t, Value value) { return 0;}

 #else

     template<class T>

     typename enable_if<typename T::value_type::LpCol,void>::type

     colLowerBound(T &t, Value value,dummy<0> = 0) {

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

         colLowerBound(*i, value);

       }

     }

     template<class T>

     typename enable_if<typename T::value_type::second_type::LpCol,

                        void>::type

     colLowerBound(T &t, Value value,dummy<1> = 1) {

       for(typename T::iterator i=t.begin();i!=t.end();++i) {

         colLowerBound(i->second, value);

       }

     }

     template<class T>

     typename enable_if<typename T::MapIt::Value::LpCol,

                        void>::type

     colLowerBound(T &t, Value value,dummy<2> = 2) {

       for(typename T::MapIt i(t); i!=INVALID; ++i){

         colLowerBound(*i, value);

       }

     }

 #endif

     /// Set the upper bound of a column (i.e a variable)

     /// The upper bound of a variable (column) has to be given by an

     /// extended number of type Value, i.e. a finite number of type

     /// Value or \ref INF.

     void colUpperBound(Col c, Value value) {

       _setColUpperBound(_cols(id(c)),value);

     };

     /// Get the upper bound of a column (i.e a variable)

     /// This function returns the upper bound for column (variable) \c c

     /// (this might be \ref INF as well).

     /// \return The upper bound for column \c c

     Value colUpperBound(Col c) const {

       return _getColUpperBound(_cols(id(c)));

     }

     ///\brief Set the upper bound of  several columns

     ///(i.e variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and applies the function on all of its elements.

     ///The upper bound of a variable (column) has to be given by an

     ///extended number of type Value, i.e. a finite number of type

     ///Value or \ref INF.

 #ifdef DOXYGEN

     template<class T>

     void colUpperBound(T &t, Value value) { return 0;}

 #else

     template<class T1>

     typename enable_if<typename T1::value_type::LpCol,void>::type

     colUpperBound(T1 &t, Value value,dummy<0> = 0) {

       for(typename T1::iterator i=t.begin();i!=t.end();++i) {

         colUpperBound(*i, value);

       }

     }

     template<class T1>

     typename enable_if<typename T1::value_type::second_type::LpCol,

                        void>::type

     colUpperBound(T1 &t, Value value,dummy<1> = 1) {

       for(typename T1::iterator i=t.begin();i!=t.end();++i) {

         colUpperBound(i->second, value);

       }

     }

     template<class T1>

     typename enable_if<typename T1::MapIt::Value::LpCol,

                        void>::type

     colUpperBound(T1 &t, Value value,dummy<2> = 2) {

       for(typename T1::MapIt i(t); i!=INVALID; ++i){

         colUpperBound(*i, value);

       }

     }

 #endif

     /// Set the lower and the upper bounds of a column (i.e a variable)

     /// The lower and the upper bounds of

     /// a variable (column) have to be given by an

     /// extended number of type Value, i.e. a finite number of type

     /// Value, -\ref INF or \ref INF.

     void colBounds(Col c, Value lower, Value upper) {

       _setColLowerBound(_cols(id(c)),lower);

       _setColUpperBound(_cols(id(c)),upper);

     }

     ///\brief Set the lower and the upper bound of several columns

     ///(i.e variables) at once

     ///

     ///This magic function takes a container as its argument

     ///and applies the function on all of its elements.

     /// The lower and the upper bounds of

     /// a variable (column) have to be given by an

     /// extended number of type Value, i.e. a finite number of type

     /// Value, -\ref INF or \ref INF.

 #ifdef DOXYGEN

     template<class T>

     void colBounds(T &t, Value lower, Value upper) { return 0;}

 #else

     template<class T2>

     typename enable_if<typename T2::value_type::LpCol,void>::type

     colBounds(T2 &t, Value lower, Value upper,dummy<0> = 0) {

       for(typename T2::iterator i=t.begin();i!=t.end();++i) {

         colBounds(*i, lower, upper);

       }

     }

     template<class T2>

     typename enable_if<typename T2::value_type::second_type::LpCol, void>::type

     colBounds(T2 &t, Value lower, Value upper,dummy<1> = 1) {

       for(typename T2::iterator i=t.begin();i!=t.end();++i) {

         colBounds(i->second, lower, upper);

       }

     }

     template<class T2>

     typename enable_if<typename T2::MapIt::Value::LpCol, void>::type

     colBounds(T2 &t, Value lower, Value upper,dummy<2> = 2) {

       for(typename T2::MapIt i(t); i!=INVALID; ++i){

         colBounds(*i, lower, upper);

       }

     }

 #endif

     /// Set the lower bound of a row (i.e a constraint)

     /// The lower bound of a constraint (row) has to be given by an

     /// extended number of type Value, i.e. a finite number of type

     /// Value or -\ref INF.

     void rowLowerBound(Row r, Value value) {

       _setRowLowerBound(_rows(id(r)),value);

     }

     /// Get the lower bound of a row (i.e a constraint)

     /// This function returns the lower bound for row (constraint) \c c

     /// (this might be -\ref INF as well).

     ///\return The lower bound for row \c r

     Value rowLowerBound(Row r) const {

       return _getRowLowerBound(_rows(id(r)));

     }

     /// Set the upper bound of a row (i.e a constraint)

     /// The upper bound of a constraint (row) has to be given by an

     /// extended number of type Value, i.e. a finite number of type

     /// Value or -\ref INF.

     void rowUpperBound(Row r, Value value) {

       _setRowUpperBound(_rows(id(r)),value);

     }

     /// Get the upper bound of a row (i.e a constraint)

     /// This function returns the upper bound for row (constraint) \c c

     /// (this might be -\ref INF as well).

     ///\return The upper bound for row \c r

     Value rowUpperBound(Row r) const {

       return _getRowUpperBound(_rows(id(r)));

     }

     ///Set an element of the objective function

     void objCoeff(Col c, Value v) {_setObjCoeff(_cols(id(c)),v); };

     ///Get an element of the objective function

     Value objCoeff(Col c) const { return _getObjCoeff(_cols(id(c))); };

     ///Set the objective function

     ///\param e is a linear expression of type \ref Expr.

     ///

     void obj(const Expr& e) {

       _setObjCoeffs(ExprIterator(e.comps.begin(), _cols),

                     ExprIterator(e.comps.end(), _cols));

       obj_const_comp = *e;

     }

     ///Get the objective function

     ///\return the objective function as a linear expression of type

     ///Expr.

     Expr obj() const {

       Expr e;

       _getObjCoeffs(InsertIterator(e.comps, _cols));

       *e = obj_const_comp;

       return e;

     }

     ///Set the direction of optimization

     void sense(Sense sense) { _setSense(sense); }

     ///Query the direction of the optimization

     Sense sense() const {return _getSense(); }

     ///Set the sense to maximization

     void max() { _setSense(MAX); }

     ///Set the sense to maximization

     void min() { _setSense(MIN); }

     ///Clear the problem

     void clear() { _clear(); _rows.clear(); _cols.clear(); }

     /// Set the message level of the solver

     void messageLevel(MessageLevel level) { _messageLevel(level); }

     /// Write the problem to a file in the given format

     /// This function writes the problem to a file in the given format.

     /// Different solver backends may support different formats.

     /// Trying to write in an unsupported format will trigger

     /// \ref UnsupportedFormatError. For the supported formats,

     /// visit the documentation of the base class of the related backends

     /// (\ref CplexBase, \ref GlpkBase etc.)

     /// \param file The file path

     /// \param format The output file format.

     void write(std::string file, std::string format = "MPS") const

     {

       _write(file.c_str(),format.c_str());

     }

     ///@}

   };

   /// Addition

   ///\relates LpBase::Expr

   ///

   inline LpBase::Expr operator+(const LpBase::Expr &a, const LpBase::Expr &b) {

     LpBase::Expr tmp(a);

     tmp+=b;

     return tmp;

   }

   ///Substraction

   ///\relates LpBase::Expr

   ///

   inline LpBase::Expr operator-(const LpBase::Expr &a, const LpBase::Expr &b) {

     LpBase::Expr tmp(a);

     tmp-=b;

     return tmp;

   }

   ///Multiply with constant

   ///\relates LpBase::Expr

   ///

   inline LpBase::Expr operator*(const LpBase::Expr &a, const LpBase::Value &b) {

     LpBase::Expr tmp(a);

     tmp*=b;

     return tmp;

   }

   ///Multiply with constant

   ///\relates LpBase::Expr

   ///

   inline LpBase::Expr operator*(const LpBase::Value &a, const LpBase::Expr &b) {

     LpBase::Expr tmp(b);

     tmp*=a;

     return tmp;

   }

   ///Divide with constant

   ///\relates LpBase::Expr

   ///

   inline LpBase::Expr operator/(const LpBase::Expr &a, const LpBase::Value &b) {

     LpBase::Expr tmp(a);

     tmp/=b;

     return tmp;

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator<=(const LpBase::Expr &e,

                                    const LpBase::Expr &f) {

     return LpBase::Constr(0, f - e, LpBase::NaN);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator<=(const LpBase::Value &e,

                                    const LpBase::Expr &f) {

     return LpBase::Constr(e, f, LpBase::NaN);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator<=(const LpBase::Expr &e,

                                    const LpBase::Value &f) {

     return LpBase::Constr(LpBase::NaN, e, f);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator>=(const LpBase::Expr &e,

                                    const LpBase::Expr &f) {

     return LpBase::Constr(0, e - f, LpBase::NaN);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator>=(const LpBase::Value &e,

                                    const LpBase::Expr &f) {

     return LpBase::Constr(LpBase::NaN, f, e);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator>=(const LpBase::Expr &e,

                                    const LpBase::Value &f) {

     return LpBase::Constr(f, e, LpBase::NaN);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator==(const LpBase::Expr &e,

                                    const LpBase::Value &f) {

     return LpBase::Constr(f, e, f);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator==(const LpBase::Expr &e,

                                    const LpBase::Expr &f) {

     return LpBase::Constr(0, f - e, 0);

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator<=(const LpBase::Value &n,

                                    const LpBase::Constr &c) {

     LpBase::Constr tmp(c);

     LEMON_ASSERT(isNaN(tmp.lowerBound()), "Wrong LP constraint");

     tmp.lowerBound()=n;

     return tmp;

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator<=(const LpBase::Constr &c,

                                    const LpBase::Value &n)

   {

     LpBase::Constr tmp(c);

     LEMON_ASSERT(isNaN(tmp.upperBound()), "Wrong LP constraint");

     tmp.upperBound()=n;

     return tmp;

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator>=(const LpBase::Value &n,

                                    const LpBase::Constr &c) {

     LpBase::Constr tmp(c);

     LEMON_ASSERT(isNaN(tmp.upperBound()), "Wrong LP constraint");

     tmp.upperBound()=n;

     return tmp;

   }

   ///Create constraint

   ///\relates LpBase::Constr

   ///

   inline LpBase::Constr operator>=(const LpBase::Constr &c,

                                    const LpBase::Value &n)

   {

     LpBase::Constr tmp(c);

     LEMON_ASSERT(isNaN(tmp.lowerBound()), "Wrong LP constraint");

     tmp.lowerBound()=n;

     return tmp;

   }

   ///Addition

   ///\relates LpBase::DualExpr

   ///

   inline LpBase::DualExpr operator+(const LpBase::DualExpr &a,

                                     const LpBase::DualExpr &b) {

     LpBase::DualExpr tmp(a);

     tmp+=b;

     return tmp;

   }

   ///Substraction

   ///\relates LpBase::DualExpr

   ///

   inline LpBase::DualExpr operator-(const LpBase::DualExpr &a,

                                     const LpBase::DualExpr &b) {

     LpBase::DualExpr tmp(a);

     tmp-=b;

     return tmp;

   }

   ///Multiply with constant

   ///\relates LpBase::DualExpr

   ///

   inline LpBase::DualExpr operator*(const LpBase::DualExpr &a,

                                     const LpBase::Value &b) {

     LpBase::DualExpr tmp(a);

     tmp*=b;

     return tmp;

   }

   ///Multiply with constant

   ///\relates LpBase::DualExpr

   ///

   inline LpBase::DualExpr operator*(const LpBase::Value &a,

                                     const LpBase::DualExpr &b) {

     LpBase::DualExpr tmp(b);

     tmp*=a;

     return tmp;

   }

   ///Divide with constant

   ///\relates LpBase::DualExpr

   ///

   inline LpBase::DualExpr operator/(const LpBase::DualExpr &a,

                                     const LpBase::Value &b) {

     LpBase::DualExpr tmp(a);

     tmp/=b;

     return tmp;

   }

   /// \ingroup lp_group

   ///

   /// \brief Common base class for LP solvers

   ///

   /// This class is an abstract base class for LP solvers. This class

   /// provides a full interface for set and modify an LP problem,

   /// solve it and retrieve the solution. You can use one of the

   /// descendants as a concrete implementation, or the \c Lp

   /// default LP solver. However, if you would like to handle LP

   /// solvers as reference or pointer in a generic way, you can use

   /// this class directly.

   class LpSolver : virtual public LpBase {

   public:

     /// The problem types for primal and dual problems

     enum ProblemType {

       /// = 0. Feasible solution hasn't been found (but may exist).

       UNDEFINED = 0,

       /// = 1. The problem has no feasible solution.

       INFEASIBLE = 1,

       /// = 2. Feasible solution found.

       FEASIBLE = 2,

       /// = 3. Optimal solution exists and found.

       OPTIMAL = 3,

       /// = 4. The cost function is unbounded.

       UNBOUNDED = 4

     };

     ///The basis status of variables

     enum VarStatus {

       /// The variable is in the basis

       BASIC,

       /// The variable is free, but not basic

       FREE,

       /// The variable has active lower bound

       LOWER,

       /// The variable has active upper bound

       UPPER,

       /// The variable is non-basic and fixed

       FIXED

     };

   protected:

     virtual SolveExitStatus _solve() = 0;

     virtual Value _getPrimal(int i) const = 0;

     virtual Value _getDual(int i) const = 0;

     virtual Value _getPrimalRay(int i) const = 0;

     virtual Value _getDualRay(int i) const = 0;

     virtual Value _getPrimalValue() const = 0;

     virtual VarStatus _getColStatus(int i) const = 0;

     virtual VarStatus _getRowStatus(int i) const = 0;

     virtual ProblemType _getPrimalType() const = 0;

     virtual ProblemType _getDualType() const = 0;

   public:

     ///Allocate a new LP problem instance

     virtual LpSolver* newSolver() const = 0;

     ///Make a copy of the LP problem

     virtual LpSolver* cloneSolver() const = 0;

     ///\name Solve the LP

     ///@{

     ///\e Solve the LP problem at hand

     ///

     ///\return The result of the optimization procedure. Possible

     ///values and their meanings can be found in the documentation of

     ///\ref SolveExitStatus.

     SolveExitStatus solve() { return _solve(); }

     ///@}

     ///\name Obtain the Solution

     ///@{

     /// The type of the primal problem

     ProblemType primalType() const {

       return _getPrimalType();

     }

     /// The type of the dual problem

     ProblemType dualType() const {

       return _getDualType();

     }

     /// Return the primal value of the column

     /// Return the primal value of the column.

     /// \pre The problem is solved.

     Value primal(Col c) const { return _getPrimal(_cols(id(c))); }

     /// Return the primal value of the expression

     /// Return the primal value of the expression, i.e. the dot

     /// product of the primal solution and the expression.

     /// \pre The problem is solved.

     Value primal(const Expr& e) const {

       double res = *e;

       for (Expr::ConstCoeffIt c(e); c != INVALID; ++c) {

         res += *c * primal(c);

       }

       return res;

     }

     /// Returns a component of the primal ray

     /// The primal ray is solution of the modified primal problem,

     /// where we change each finite bound to 0, and we looking for a

     /// negative objective value in case of minimization, and positive

     /// objective value for maximization. If there is such solution,

     /// that proofs the unsolvability of the dual problem, and if a

     /// feasible primal solution exists, then the unboundness of

     /// primal problem.

     ///

     /// \pre The problem is solved and the dual problem is infeasible.

     /// \note Some solvers does not provide primal ray calculation

     /// functions.

     Value primalRay(Col c) const { return _getPrimalRay(_cols(id(c))); }

     /// Return the dual value of the row

     /// Return the dual value of the row.

     /// \pre The problem is solved.

     Value dual(Row r) const { return _getDual(_rows(id(r))); }

     /// Return the dual value of the dual expression

     /// Return the dual value of the dual expression, i.e. the dot

     /// product of the dual solution and the dual expression.

     /// \pre The problem is solved.

     Value dual(const DualExpr& e) const {

       double res = 0.0;

       for (DualExpr::ConstCoeffIt r(e); r != INVALID; ++r) {

         res += *r * dual(r);

       }

       return res;

     }

     /// Returns a component of the dual ray

     /// The dual ray is solution of the modified primal problem, where

     /// we change each finite bound to 0 (i.e. the objective function

     /// coefficients in the primal problem), and we looking for a

     /// ositive objective value. If there is such solution, that

     /// proofs the unsolvability of the primal problem, and if a

     /// feasible dual solution exists, then the unboundness of

     /// dual problem.

     ///

     /// \pre The problem is solved and the primal problem is infeasible.

     /// \note Some solvers does not provide dual ray calculation

     /// functions.

     Value dualRay(Row r) const { return _getDualRay(_rows(id(r))); }

     /// Return the basis status of the column

     /// \see VarStatus

     VarStatus colStatus(Col c) const { return _getColStatus(_cols(id(c))); }

     /// Return the basis status of the row

     /// \see VarStatus

     VarStatus rowStatus(Row r) const { return _getRowStatus(_rows(id(r))); }

     ///The value of the objective function

     ///\return

     ///- \ref INF or -\ref INF means either infeasibility or unboundedness

     /// of the primal problem, depending on whether we minimize or maximize.

     ///- \ref NaN if no primal solution is found.

     ///- The (finite) objective value if an optimal solution is found.

     Value primal() const { return _getPrimalValue()+obj_const_comp;}

     ///@}

   protected:

   };

   /// \ingroup lp_group

   ///

   /// \brief Common base class for MIP solvers

   ///

   /// This class is an abstract base class for MIP solvers. This class

   /// provides a full interface for set and modify an MIP problem,

   /// solve it and retrieve the solution. You can use one of the

   /// descendants as a concrete implementation, or the \c Lp

   /// default MIP solver. However, if you would like to handle MIP

   /// solvers as reference or pointer in a generic way, you can use

   /// this class directly.

   class MipSolver : virtual public LpBase {

   public:

     /// The problem types for MIP problems

     enum ProblemType {

       /// = 0. Feasible solution hasn't been found (but may exist).

       UNDEFINED = 0,

       /// = 1. The problem has no feasible solution.

       INFEASIBLE = 1,

       /// = 2. Feasible solution found.

       FEASIBLE = 2,

       /// = 3. Optimal solution exists and found.

       OPTIMAL = 3,

       /// = 4. The cost function is unbounded.

       ///The Mip or at least the relaxed problem is unbounded.

       UNBOUNDED = 4

     };

     ///Allocate a new MIP problem instance

     virtual MipSolver* newSolver() const = 0;

     ///Make a copy of the MIP problem

     virtual MipSolver* cloneSolver() const = 0;

     ///\name Solve the MIP

     ///@{

     /// Solve the MIP problem at hand

     ///

     ///\return The result of the optimization procedure. Possible

     ///values and their meanings can be found in the documentation of

     ///\ref SolveExitStatus.

     SolveExitStatus solve() { return _solve(); }

     ///@}

     ///\name Set Column Type

     ///@{

     ///Possible variable (column) types (e.g. real, integer, binary etc.)

     enum ColTypes {

       /// = 0. Continuous variable (default).

       REAL = 0,

       /// = 1. Integer variable.

       INTEGER = 1

     };

     ///Sets the type of the given column to the given type

     ///Sets the type of the given column to the given type.

     ///

     void colType(Col c, ColTypes col_type) {

       _setColType(_cols(id(c)),col_type);

     }

     ///Gives back the type of the column.

     ///Gives back the type of the column.

     ///

     ColTypes colType(Col c) const {

       return _getColType(_cols(id(c)));

     }

     ///@}

     ///\name Obtain the Solution

     ///@{

     /// The type of the MIP problem

     ProblemType type() const {

       return _getType();

     }

     /// Return the value of the row in the solution

     ///  Return the value of the row in the solution.

     /// \pre The problem is solved.

     Value sol(Col c) const { return _getSol(_cols(id(c))); }

     /// Return the value of the expression in the solution

     /// Return the value of the expression in the solution, i.e. the

     /// dot product of the solution and the expression.

     /// \pre The problem is solved.

     Value sol(const Expr& e) const {

       double res = *e;

       for (Expr::ConstCoeffIt c(e); c != INVALID; ++c) {

         res += *c * sol(c);

       }

       return res;

     }

     ///The value of the objective function

     ///\return

     ///- \ref INF or -\ref INF means either infeasibility or unboundedness

     /// of the problem, depending on whether we minimize or maximize.

     ///- \ref NaN if no primal solution is found.

     ///- The (finite) objective value if an optimal solution is found.

     Value solValue() const { return _getSolValue()+obj_const_comp;}

     ///@}

   protected:

     virtual SolveExitStatus _solve() = 0;

     virtual ColTypes _getColType(int col) const = 0;

     virtual void _setColType(int col, ColTypes col_type) = 0;

     virtual ProblemType _getType() const = 0;

     virtual Value _getSol(int i) const = 0;

     virtual Value _getSolValue() const = 0;

   };

 } //namespace lemon

 #endif //LEMON_LP_BASE_H