1 | /* -*- mode: C++; indent-tabs-mode: nil; -*- |
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2 | * |
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3 | * This file is a part of LEMON, a generic C++ optimization library. |
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4 | * |
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5 | * Copyright (C) 2003-2010 |
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6 | * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport |
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7 | * (Egervary Research Group on Combinatorial Optimization, EGRES). |
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8 | * |
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9 | * Permission to use, modify and distribute this software is granted |
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10 | * provided that this copyright notice appears in all copies. For |
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11 | * precise terms see the accompanying LICENSE file. |
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12 | * |
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13 | * This software is provided "AS IS" with no warranty of any kind, |
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14 | * express or implied, and with no claim as to its suitability for any |
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15 | * purpose. |
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16 | * |
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17 | */ |
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18 | |
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19 | namespace lemon { |
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20 | /** |
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21 | [PAGE]sec_lp[PAGE] Linear Programming Interface |
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22 | |
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23 | Linear programming (LP) is one of the most important general methods of |
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24 | operations research. Countless optimization problems can be formulated |
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25 | and solved using LP techniques. |
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26 | Therefore, developing efficient LP solvers has been of high practical |
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27 | interest for a long time. |
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28 | Nowadays various efficient LP solvers are available, including both |
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29 | open source and commercial software packages. |
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30 | Therefore, LEMON does not implement its own solver, but it features |
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31 | wrapper classes for several known LP packages providing a common |
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32 | high-level interface for all of them. |
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33 | |
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34 | The advantage of this approach is twofold. First, our C++ interface is |
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35 | more comfortable than the typical native interfaces of the solvers. |
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36 | Second, changing the underlying solver in a certain application using |
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37 | LEMON's LP interface needs no effort. So, for example, one may try her |
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38 | idea using an open source solver, demonstrate its usability for a customer |
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39 | and if it works well, but the performance should be improved, then the |
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40 | customer may decide to purchase and use a better commercial solver. |
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41 | |
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42 | Currently, the following linear and mixed integer programming packages are |
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43 | supported: GLPK, Clp, Cbc, ILOG CPLEX and SoPlex. |
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44 | However, additional wrapper classes for new solvers can also be implemented |
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45 | quite easily. |
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46 | |
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47 | [SEC]sec_lp_basics[SEC] Basic LP Features |
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48 | |
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49 | The following example demonstrates how simple it is to formalize and solve |
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50 | an LP problem in LEMON. You can find the whole program in \ref lp_demo.cc. |
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51 | |
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52 | \dontinclude lp_demo.cc |
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53 | \skip Create |
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54 | \until } |
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55 | \until } |
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56 | |
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57 | \ref LpBase::Col "Lp::Col" type represents the columns, i.e. the (primal) |
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58 | variables of the LP problem. The numerical operators can be used to form |
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59 | expressions from columns, which are required for specifying rows |
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60 | (constraints) and the objective function. Due to the suitable operator |
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61 | overloads, a problem can be described in C++ conveniently, directly as it is |
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62 | expressed in mathematics. |
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63 | After specifying all the parameters of the problem, we can solve it using |
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64 | the \ref LpSolver::solve() "solve()" function. The status of the solution |
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65 | (namely OPTIMAL, UNBOUNDED, INFEASIBLE etc.) can be obtained using |
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66 | \ref LpSolver::primalType() "primalType()" and the results can be |
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67 | obtained using \ref LpSolver::primal() "primal()". |
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68 | |
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69 | The above problem has an optimal solution. If you execute this example, |
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70 | it will print out the following results. |
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71 | |
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72 | \code |
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73 | Objective function value: 67.5 |
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74 | x1 = 7.5 |
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75 | x2 = 10 |
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76 | \endcode |
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77 | |
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78 | However, if you, for example, removed the lines that specify the rows, |
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79 | you would obtain an LP problem, for which the objective function value |
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80 | is unbounded over the feasible solutions. Thus the program would print |
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81 | out this line. |
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82 | |
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83 | \code |
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84 | Optimal solution found. |
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85 | \endcode |
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86 | |
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87 | If you would like to build linear expressions or constraints in more steps, |
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88 | then you can use the classes \ref LpBase::Expr "Lp::Expr" and |
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89 | \ref LpBase::Constr "Lp::Constr". For example, this line |
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90 | \code |
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91 | lp.addRow(x1 - 5 <= x2); |
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92 | \endcode |
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93 | can also be written as |
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94 | \code |
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95 | Lp::Expr e1, e2; |
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96 | e1 = x1; |
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97 | e1 -= 5; |
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98 | e2 = x2; |
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99 | Lp::Constr c = e1 <= e2; |
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100 | lp.addRow(c); |
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101 | \endcode |
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102 | |
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103 | These classes make it easy to build more complex expressions and constraints, |
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104 | e.g. using loops. For example, we can sum all the variables (columns) in an |
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105 | expression using the \ref LpBase::ColIt "Lp::ColIt" iterator, which can be |
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106 | used the same as node and arc iterators for graphs. |
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107 | |
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108 | \code |
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109 | Lp::Expr sum; |
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110 | for (Lp::ColIt col(lp); col != INVALID; ++col) { |
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111 | sum += col; |
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112 | } |
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113 | \endcode |
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114 | |
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115 | After solving the problem, you can query the value of any primal expression, |
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116 | not just the individual columns and the objective function. |
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117 | |
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118 | \todo Check this example after closing ticket #326. |
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119 | |
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120 | \code |
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121 | std::cout << lp.primal(sum) << std::endl; |
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122 | \endcode |
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123 | |
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124 | Of course, working with the dual problem is also supported by the LP |
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125 | interface. \ref LpBase::Row "Lp::Row" represents the rows, i.e. the dual |
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126 | variables of the LP problem. |
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127 | |
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128 | \code |
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129 | Lp::Row r = lp.addRow(x2 >= 3); |
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130 | \endcode |
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131 | |
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132 | The dual solutions of an LP problem can be obtained using the functions |
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133 | \ref LpSolver::dualType() "dualType()" and \ref LpSolver::dual "dual()" |
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134 | (after solving the problem). |
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135 | |
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136 | \code |
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137 | std::cout << lp.dual(r) << std::endl; |
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138 | \endcode |
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139 | |
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140 | \ref LpBase::DualExpr "Lp::DualExpr" can be used to build dual expressions |
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141 | from rows and \ref LpBase::RowIt "Lp::RowIt" can be used to list the rows. |
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142 | |
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143 | \code |
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144 | Lp::DualExpr dual_sum; |
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145 | for (Lp::RowIt row(lp); row != INVALID; ++row) { |
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146 | dual_sum += row; |
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147 | } |
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148 | \endcode |
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149 | |
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150 | The LP solver interface provides several other features, which are |
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151 | documented in the classes \ref LpBase and \ref LpSolver. |
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152 | For detailed documentation, see these classes in the reference manual. |
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153 | |
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154 | If you would like to use a specific solver instead of the default one, |
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155 | you can use \ref GlpkLp, \ref ClpLp, \ref CplexLp etc. instead of |
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156 | the class \ref Lp. |
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157 | |
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158 | |
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159 | [SEC]sec_lp_mip[SEC] Mixed Integer Programming |
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160 | |
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161 | LEMON also provides a similar high-level interface for mixed integer |
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162 | programming (MIP) solvers. The default MIP solver can be used through |
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163 | the class \ref Mip, while the concrete solvers are implemented in the |
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164 | classes \ref GlpkMip, \ref CbcMip, \ref CplexMip etc. |
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165 | |
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166 | The following example demonstrates the usage of the MIP solver interface. |
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167 | The whole program can be found in \ref mip_demo.cc. |
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168 | |
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169 | \dontinclude mip_demo.cc |
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170 | \skip Create |
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171 | \until } |
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172 | \until } |
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173 | |
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174 | \todo Check this demo file after closing ticket #326. |
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175 | E.g. could we write <tt>mip.sol()</tt> instead of <tt>mip.solValue()</tt>? |
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176 | Or could we write <tt>primalValue()</tt> for \c Lp in \c lp_demo.cc? |
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177 | |
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178 | In this program, the same problem is built as in the above example, |
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179 | with the exception that the variable \c x1 is specified to be |
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180 | \c INTEGER valued. \c x2 is set to \c REAL, which is the default option. |
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181 | |
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182 | \code |
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183 | // Set the type of the columns |
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184 | mip.colType(x1, Mip::INTEGER); |
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185 | mip.colType(x2, Mip::REAL); |
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186 | \endcode |
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187 | |
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188 | Because of this integrality requirement, the results will be different |
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189 | from the LP solution. |
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190 | |
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191 | \code |
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192 | Objective function value: 67 |
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193 | x1 = 8 |
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194 | x2 = 9 |
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195 | \endcode |
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196 | |
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197 | The documnetation of the MIP solver interface can be found in the |
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198 | reference manual at the class \ref MipSolver. The common parts of the |
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199 | LP and MIP interfaces are docmented in their common ancestor class |
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200 | \ref LpBase. |
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201 | |
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202 | |
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203 | [SEC]sec_lp_optimization[SEC] Solving Complex Optimization Problems |
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204 | |
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205 | The LP and MIP solvers are powerful tools for solving various complex |
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206 | optimization problems, as well. |
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207 | The following example solves a maximum flow problem with linear |
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208 | programming, see the whole program in \re lp_maxflow_demo.cc. |
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209 | Several other graph optimization problems can also be expressed as |
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210 | linear programs and the interface provided in LEMON facilitates solving |
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211 | them easily (though usually not so efficiently as by a direct |
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212 | combinatorial method, if one exists). |
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213 | |
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214 | \dontinclude lp_maxflow_demo.cc |
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215 | \skip DIGRAPH_TYPEDEFS |
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216 | \until solve() |
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217 | |
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218 | \note This problem can be solved much more efficiently using common |
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219 | combinatorial optimization methods, such as the \ref Preflow |
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220 | "preflow push-relabel algorithm". |
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221 | |
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222 | [TRAILER] |
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223 | */ |
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224 | } |
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