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1 ## Copyright (C) 2005, 2006, 2007 Nicolo' Giorgetti |
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2 ## |
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3 ## This file is part of Octave. |
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4 ## |
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5 ## Octave is free software; you can redistribute it and/or modify it |
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6 ## under the terms of the GNU General Public License as published by |
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7 ## the Free Software Foundation; either version 3 of the License, or (at |
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8 ## your option) any later version. |
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9 ## |
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10 ## Octave is distributed in the hope that it will be useful, but |
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU |
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13 ## General Public License for more details. |
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14 ## |
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15 ## You should have received a copy of the GNU General Public License |
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16 ## along with Octave; see the file COPYING. If not, see |
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17 ## <http://www.gnu.org/licenses/>. |
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18 |
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19 ## -*- texinfo -*- |
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20 ## @deftypefn {Function File} {[@var{xopt}, @var{fmin}, @var{status}, @var{extra}] =} glpk (@var{c}, @var{a}, @var{b}, @var{lb}, @var{ub}, @var{ctype}, @var{vartype}, @var{sense}, @var{param}) |
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21 ## Solve a linear program using the GNU GLPK library. Given three |
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22 ## arguments, @code{glpk} solves the following standard LP: |
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23 ## |
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24 ## @iftex |
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25 ## @tex |
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26 ## $$ |
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27 ## \min_x C^T x |
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28 ## $$ |
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29 ## @end tex |
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30 ## @end iftex |
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31 ## @ifnottex |
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32 ## @example |
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33 ## min C'*x |
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34 ## @end example |
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35 ## @end ifnottex |
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36 ## |
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37 ## subject to |
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38 ## |
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39 ## @iftex |
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40 ## @tex |
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41 ## $$ |
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42 ## Ax = b \qquad x \geq 0 |
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43 ## $$ |
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44 ## @end tex |
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45 ## @end iftex |
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46 ## @ifnottex |
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47 ## @example |
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48 ## @group |
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49 ## A*x = b |
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50 ## x >= 0 |
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51 ## @end group |
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52 ## @end example |
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53 ## @end ifnottex |
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54 ## |
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55 ## but may also solve problems of the form |
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56 ## |
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57 ## @iftex |
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58 ## @tex |
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59 ## $$ |
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60 ## [ \min_x | \max_x ] C^T x |
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61 ## $$ |
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62 ## @end tex |
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63 ## @end iftex |
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64 ## @ifnottex |
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65 ## @example |
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66 ## [ min | max ] C'*x |
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67 ## @end example |
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68 ## @end ifnottex |
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69 ## |
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70 ## subject to |
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71 ## |
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72 ## @iftex |
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73 ## @tex |
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74 ## $$ |
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75 ## Ax [ = | \leq | \geq ] b \qquad LB \leq x \leq UB |
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76 ## $$ |
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77 ## @end tex |
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78 ## @end iftex |
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79 ## @ifnottex |
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80 ## @example |
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81 ## @group |
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82 ## A*x [ "=" | "<=" | ">=" ] b |
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83 ## x >= LB |
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84 ## x <= UB |
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85 ## @end group |
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86 ## @end example |
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87 ## @end ifnottex |
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88 ## |
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89 ## Input arguments: |
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90 ## |
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91 ## @table @var |
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92 ## @item c |
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93 ## A column array containing the objective function coefficients. |
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94 ## |
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95 ## @item a |
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96 ## A matrix containing the constraints coefficients. |
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97 ## |
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98 ## @item b |
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99 ## A column array containing the right-hand side value for each constraint |
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100 ## in the constraint matrix. |
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101 ## |
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102 ## @item lb |
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103 ## An array containing the lower bound on each of the variables. If |
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104 ## @var{lb} is not supplied, the default lower bound for the variables is |
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105 ## zero. |
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106 ## |
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107 ## @item ub |
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108 ## An array containing the upper bound on each of the variables. If |
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109 ## @var{ub} is not supplied, the default upper bound is assumed to be |
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110 ## infinite. |
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111 ## |
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112 ## @item ctype |
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113 ## An array of characters containing the sense of each constraint in the |
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114 ## constraint matrix. Each element of the array may be one of the |
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115 ## following values |
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116 ## @table @code |
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117 ## @item "F" |
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118 ## A free (unbounded) constraint (the constraint is ignored). |
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119 ## @item "U" |
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120 ## An inequality constraint with an upper bound (@code{A(i,:)*x <= b(i)}). |
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121 ## @item "S" |
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122 ## An equality constraint (@code{A(i,:)*x = b(i)}). |
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123 ## @item "L" |
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124 ## An inequality with a lower bound (@code{A(i,:)*x >= b(i)}). |
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125 ## @item "D" |
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126 ## An inequality constraint with both upper and lower bounds |
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127 ## (@code{A(i,:)*x >= -b(i)} @emph{and} (@code{A(i,:)*x <= b(i)}). |
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128 ## @end table |
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129 ## |
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130 ## @item vartype |
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131 ## A column array containing the types of the variables. |
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132 ## @table @code |
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133 ## @item "C" |
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134 ## A continuous variable. |
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135 ## @item "I" |
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136 ## An integer variable. |
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137 ## @end table |
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138 ## |
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139 ## @item sense |
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140 ## If @var{sense} is 1, the problem is a minimization. If @var{sense} is |
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141 ## -1, the problem is a maximization. The default value is 1. |
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142 ## |
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143 ## @item param |
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144 ## A structure containing the following parameters used to define the |
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145 ## behavior of solver. Missing elements in the structure take on default |
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146 ## values, so you only need to set the elements that you wish to change |
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147 ## from the default. |
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148 ## |
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149 ## Integer parameters: |
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150 ## |
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151 ## @table @code |
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152 ## @item msglev (@code{LPX_K_MSGLEV}, default: 1) |
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153 ## Level of messages output by solver routines: |
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154 ## @table @asis |
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155 ## @item 0 |
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156 ## No output. |
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157 ## @item 1 |
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158 ## Error messages only. |
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159 ## @item 2 |
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160 ## Normal output . |
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161 ## @item 3 |
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162 ## Full output (includes informational messages). |
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163 ## @end table |
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164 ## |
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165 ## @item scale (@code{LPX_K_SCALE}, default: 1) |
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166 ## Scaling option: |
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167 ## @table @asis |
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168 ## @item 0 |
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169 ## No scaling. |
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170 ## @item 1 |
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171 ## Equilibration scaling. |
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172 ## @item 2 |
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173 ## Geometric mean scaling, then equilibration scaling. |
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174 ## @end table |
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175 ## |
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176 ## @item dual (@code{LPX_K_DUAL}, default: 0) |
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177 ## Dual simplex option: |
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178 ## @table @asis |
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179 ## @item 0 |
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180 ## Do not use the dual simplex. |
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181 ## @item 1 |
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182 ## If initial basic solution is dual feasible, use the dual simplex. |
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183 ## @end table |
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184 ## |
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185 ## @item price (@code{LPX_K_PRICE}, default: 1) |
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186 ## Pricing option (for both primal and dual simplex): |
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187 ## @table @asis |
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188 ## @item 0 |
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189 ## Textbook pricing. |
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190 ## @item 1 |
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191 ## Steepest edge pricing. |
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192 ## @end table |
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193 ## |
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194 ## @item round (@code{LPX_K_ROUND}, default: 0) |
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195 ## Solution rounding option: |
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196 ## @table @asis |
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197 ## @item 0 |
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198 ## Report all primal and dual values "as is". |
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199 ## @item 1 |
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200 ## Replace tiny primal and dual values by exact zero. |
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201 ## @end table |
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202 ## |
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203 ## @item itlim (@code{LPX_K_ITLIM}, default: -1) |
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204 ## Simplex iterations limit. If this value is positive, it is decreased by |
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205 ## one each time when one simplex iteration has been performed, and |
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206 ## reaching zero value signals the solver to stop the search. Negative |
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207 ## value means no iterations limit. |
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208 ## |
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209 ## @item itcnt (@code{LPX_K_OUTFRQ}, default: 200) |
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210 ## Output frequency, in iterations. This parameter specifies how |
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211 ## frequently the solver sends information about the solution to the |
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212 ## standard output. |
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213 ## |
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214 ## @item branch (@code{LPX_K_BRANCH}, default: 2) |
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215 ## Branching heuristic option (for MIP only): |
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216 ## @table @asis |
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217 ## @item 0 |
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218 ## Branch on the first variable. |
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219 ## @item 1 |
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220 ## Branch on the last variable. |
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221 ## @item 2 |
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222 ## Branch using a heuristic by Driebeck and Tomlin. |
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223 ## @end table |
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224 ## |
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225 ## @item btrack (@code{LPX_K_BTRACK}, default: 2) |
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226 ## Backtracking heuristic option (for MIP only): |
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227 ## @table @asis |
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228 ## @item 0 |
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229 ## Depth first search. |
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230 ## @item 1 |
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231 ## Breadth first search. |
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232 ## @item 2 |
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233 ## Backtrack using the best projection heuristic. |
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234 ## @end table |
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235 ## |
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236 ## @item presol (@code{LPX_K_PRESOL}, default: 1) |
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237 ## If this flag is set, the routine lpx_simplex solves the problem using |
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238 ## the built-in LP presolver. Otherwise the LP presolver is not used. |
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239 ## |
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240 ## @item lpsolver (default: 1) |
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241 ## Select which solver to use. If the problem is a MIP problem this flag |
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242 ## will be ignored. |
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243 ## @table @asis |
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244 ## @item 1 |
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245 ## Revised simplex method. |
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246 ## @item 2 |
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247 ## Interior point method. |
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248 ## @end table |
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249 ## @item save (default: 0) |
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250 ## If this parameter is nonzero, save a copy of the problem in |
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251 ## CPLEX LP format to the file @file{"outpb.lp"}. There is currently no |
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252 ## way to change the name of the output file. |
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253 ## @end table |
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254 ## |
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255 ## Real parameters: |
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256 ## |
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257 ## @table @code |
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258 ## @item relax (@code{LPX_K_RELAX}, default: 0.07) |
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259 ## Relaxation parameter used in the ratio test. If it is zero, the textbook |
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260 ## ratio test is used. If it is non-zero (should be positive), Harris' |
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261 ## two-pass ratio test is used. In the latter case on the first pass of the |
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262 ## ratio test basic variables (in the case of primal simplex) or reduced |
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263 ## costs of non-basic variables (in the case of dual simplex) are allowed |
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264 ## to slightly violate their bounds, but not more than |
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265 ## @code{relax*tolbnd} or @code{relax*toldj (thus, @code{relax} is a |
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266 ## percentage of @code{tolbnd} or @code{toldj}}. |
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267 ## |
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268 ## @item tolbnd (@code{LPX_K_TOLBND}, default: 10e-7) |
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269 ## Relative tolerance used to check if the current basic solution is primal |
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270 ## feasible. It is not recommended that you change this parameter unless you |
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271 ## have a detailed understanding of its purpose. |
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272 ## |
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273 ## @item toldj (@code{LPX_K_TOLDJ}, default: 10e-7) |
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274 ## Absolute tolerance used to check if the current basic solution is dual |
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275 ## feasible. It is not recommended that you change this parameter unless you |
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276 ## have a detailed understanding of its purpose. |
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277 ## |
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278 ## @item tolpiv (@code{LPX_K_TOLPIV}, default: 10e-9) |
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279 ## Relative tolerance used to choose eligible pivotal elements of the |
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280 ## simplex table. It is not recommended that you change this parameter unless you |
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281 ## have a detailed understanding of its purpose. |
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282 ## |
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283 ## @item objll (@code{LPX_K_OBJLL}, default: -DBL_MAX) |
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284 ## Lower limit of the objective function. If on the phase II the objective |
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285 ## function reaches this limit and continues decreasing, the solver stops |
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286 ## the search. This parameter is used in the dual simplex method only. |
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287 ## |
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288 ## @item objul (@code{LPX_K_OBJUL}, default: +DBL_MAX) |
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289 ## Upper limit of the objective function. If on the phase II the objective |
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290 ## function reaches this limit and continues increasing, the solver stops |
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291 ## the search. This parameter is used in the dual simplex only. |
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292 ## |
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293 ## @item tmlim (@code{LPX_K_TMLIM}, default: -1.0) |
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294 ## Searching time limit, in seconds. If this value is positive, it is |
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295 ## decreased each time when one simplex iteration has been performed by the |
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296 ## amount of time spent for the iteration, and reaching zero value signals |
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297 ## the solver to stop the search. Negative value means no time limit. |
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298 ## |
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299 ## @item outdly (@code{LPX_K_OUTDLY}, default: 0.0) |
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300 ## Output delay, in seconds. This parameter specifies how long the solver |
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301 ## should delay sending information about the solution to the standard |
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302 ## output. Non-positive value means no delay. |
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303 ## |
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304 ## @item tolint (@code{LPX_K_TOLINT}, default: 10e-5) |
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305 ## Relative tolerance used to check if the current basic solution is integer |
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306 ## feasible. It is not recommended that you change this parameter unless |
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307 ## you have a detailed understanding of its purpose. |
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308 ## |
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309 ## @item tolobj (@code{LPX_K_TOLOBJ}, default: 10e-7) |
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310 ## Relative tolerance used to check if the value of the objective function |
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311 ## is not better than in the best known integer feasible solution. It is |
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312 ## not recommended that you change this parameter unless you have a |
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313 ## detailed understanding of its purpose. |
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314 ## @end table |
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315 ## @end table |
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316 ## |
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317 ## Output values: |
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318 ## |
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319 ## @table @var |
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320 ## @item xopt |
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321 ## The optimizer (the value of the decision variables at the optimum). |
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322 ## @item fopt |
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323 ## The optimum value of the objective function. |
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324 ## @item status |
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325 ## Status of the optimization. |
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326 ## |
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327 ## Simplex Method: |
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328 ## @table @asis |
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329 ## @item 180 (@code{LPX_OPT}) |
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330 ## Solution is optimal. |
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331 ## @item 181 (@code{LPX_FEAS}) |
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332 ## Solution is feasible. |
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333 ## @item 182 (@code{LPX_INFEAS}) |
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334 ## Solution is infeasible. |
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335 ## @item 183 (@code{LPX_NOFEAS}) |
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336 ## Problem has no feasible solution. |
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337 ## @item 184 (@code{LPX_UNBND}) |
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338 ## Problem has no unbounded solution. |
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339 ## @item 185 (@code{LPX_UNDEF}) |
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340 ## Solution status is undefined. |
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341 ## @end table |
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342 ## Interior Point Method: |
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343 ## @table @asis |
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344 ## @item 150 (@code{LPX_T_UNDEF}) |
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345 ## The interior point method is undefined. |
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346 ## @item 151 (@code{LPX_T_OPT}) |
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347 ## The interior point method is optimal. |
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348 ## @end table |
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349 ## Mixed Integer Method: |
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350 ## @table @asis |
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351 ## @item 170 (@code{LPX_I_UNDEF}) |
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352 ## The status is undefined. |
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353 ## @item 171 (@code{LPX_I_OPT}) |
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354 ## The solution is integer optimal. |
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355 ## @item 172 (@code{LPX_I_FEAS}) |
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356 ## Solution integer feasible but its optimality has not been proven |
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357 ## @item 173 (@code{LPX_I_NOFEAS}) |
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358 ## No integer feasible solution. |
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359 ## @end table |
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360 ## @noindent |
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361 ## If an error occurs, @var{status} will contain one of the following |
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362 ## codes: |
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363 ## |
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364 ## @table @asis |
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365 ## @item 204 (@code{LPX_E_FAULT}) |
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366 ## Unable to start the search. |
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367 ## @item 205 (@code{LPX_E_OBJLL}) |
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368 ## Objective function lower limit reached. |
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369 ## @item 206 (@code{LPX_E_OBJUL}) |
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370 ## Objective function upper limit reached. |
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371 ## @item 207 (@code{LPX_E_ITLIM}) |
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372 ## Iterations limit exhausted. |
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373 ## @item 208 (@code{LPX_E_TMLIM}) |
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374 ## Time limit exhausted. |
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375 ## @item 209 (@code{LPX_E_NOFEAS}) |
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376 ## No feasible solution. |
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377 ## @item 210 (@code{LPX_E_INSTAB}) |
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378 ## Numerical instability. |
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379 ## @item 211 (@code{LPX_E_SING}) |
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380 ## Problems with basis matrix. |
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381 ## @item 212 (@code{LPX_E_NOCONV}) |
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382 ## No convergence (interior). |
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383 ## @item 213 (@code{LPX_E_NOPFS}) |
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384 ## No primal feasible solution (LP presolver). |
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385 ## @item 214 (@code{LPX_E_NODFS}) |
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386 ## No dual feasible solution (LP presolver). |
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387 ## @end table |
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388 ## @item extra |
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389 ## A data structure containing the following fields: |
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390 ## @table @code |
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391 ## @item lambda |
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392 ## Dual variables. |
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393 ## @item redcosts |
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394 ## Reduced Costs. |
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395 ## @item time |
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396 ## Time (in seconds) used for solving LP/MIP problem. |
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397 ## @item mem |
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398 ## Memory (in bytes) used for solving LP/MIP problem (this is not |
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399 ## available if the version of GLPK is 4.15 or later). |
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400 ## @end table |
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401 ## @end table |
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402 ## |
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403 ## Example: |
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404 ## |
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405 ## @example |
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406 ## @group |
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407 ## c = [10, 6, 4]'; |
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408 ## a = [ 1, 1, 1; |
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409 ## 10, 4, 5; |
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410 ## 2, 2, 6]; |
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411 ## b = [100, 600, 300]'; |
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412 ## lb = [0, 0, 0]'; |
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413 ## ub = []; |
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414 ## ctype = "UUU"; |
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415 ## vartype = "CCC"; |
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416 ## s = -1; |
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417 ## |
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418 ## param.msglev = 1; |
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419 ## param.itlim = 100; |
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420 ## |
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421 ## [xmin, fmin, status, extra] = ... |
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422 ## glpk (c, a, b, lb, ub, ctype, vartype, s, param); |
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423 ## @end group |
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424 ## @end example |
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425 ## @end deftypefn |
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426 |
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427 ## Author: Nicolo' Giorgetti <giorgetti@dii.unisi.it> |
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428 ## Adapted-by: jwe |
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429 |
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430 function [xopt, fmin, status, extra] = glpk (c, a, b, lb, ub, ctype, vartype, sense, param) |
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431 |
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5233
|
432 ## If there is no input output the version and syntax |
|
5237
|
433 if (nargin < 3 || nargin > 9) |
|
6046
|
434 print_usage (); |
|
5233
|
435 return; |
|
|
436 endif |
|
5232
|
437 |
|
5233
|
438 if (all (size (c) > 1) || iscomplex (c) || ischar (c)) |
|
|
439 error ("C must be a real vector"); |
|
|
440 return; |
|
|
441 endif |
|
|
442 nx = length (c); |
|
|
443 ## Force column vector. |
|
|
444 c = c(:); |
|
5232
|
445 |
|
5237
|
446 ## 2) Matrix constraint |
|
5232
|
447 |
|
5233
|
448 if (isempty (a)) |
|
|
449 error ("A cannot be an empty matrix"); |
|
|
450 return; |
|
|
451 endif |
|
|
452 [nc, nxa] = size(a); |
|
|
453 if (! isreal (a) || nxa != nx) |
|
|
454 error ("A must be a real valued %d by %d matrix", nc, nx); |
|
|
455 return; |
|
|
456 endif |
|
|
457 |
|
5237
|
458 ## 3) RHS |
|
5232
|
459 |
|
5233
|
460 if (isempty (b)) |
|
|
461 error ("B cannot be an empty vector"); |
|
|
462 return; |
|
|
463 endif |
|
|
464 if (! isreal (b) || length (b) != nc) |
|
|
465 error ("B must be a real valued %d by 1 vector", nc); |
|
|
466 return; |
|
|
467 endif |
|
|
468 |
|
5237
|
469 ## 4) Vector with the lower bound of each variable |
|
5232
|
470 |
|
5237
|
471 if (nargin > 3) |
|
5233
|
472 if (isempty (lb)) |
|
5237
|
473 lb = zeros (0, nx, 1); |
|
5233
|
474 elseif (! isreal (lb) || all (size (lb) > 1) || length (lb) != nx) |
|
|
475 error ("LB must be a real valued %d by 1 column vector", nx); |
|
|
476 return; |
|
|
477 endif |
|
|
478 else |
|
5237
|
479 lb = zeros (nx, 1); |
|
7151
|
480 endif |
|
5233
|
481 |
|
5237
|
482 ## 5) Vector with the upper bound of each variable |
|
5232
|
483 |
|
5237
|
484 if (nargin > 4) |
|
5233
|
485 if (isempty (ub)) |
|
|
486 ub = repmat (Inf, nx, 1); |
|
|
487 elseif (! isreal (ub) || all (size (ub) > 1) || length (ub) != nx) |
|
|
488 error ("UB must be a real valued %d by 1 column vector", nx); |
|
|
489 return; |
|
|
490 endif |
|
|
491 else |
|
|
492 ub = repmat (Inf, nx, 1); |
|
7151
|
493 endif |
|
5232
|
494 |
|
5237
|
495 ## 6) Sense of each constraint |
|
5232
|
496 |
|
5237
|
497 if (nargin > 5) |
|
|
498 if (isempty (ctype)) |
|
|
499 ctype = repmat ("S", nc, 1); |
|
|
500 elseif (! ischar (ctype) || all (size (ctype) > 1) || length (ctype) != nc) |
|
5244
|
501 error ("CTYPE must be a char valued vector of length %d", nc); |
|
5237
|
502 return; |
|
|
503 elseif (! all (ctype == "F" | ctype == "U" | ctype == "S" |
|
|
504 | ctype == "L" | ctype == "D")) |
|
|
505 error ("CTYPE must contain only F, U, S, L, or D"); |
|
|
506 return; |
|
|
507 endif |
|
|
508 else |
|
|
509 ctype = repmat ("S", nc, 1); |
|
7151
|
510 endif |
|
5237
|
511 |
|
|
512 ## 7) Vector with the type of variables |
|
|
513 |
|
|
514 if (nargin > 6) |
|
5289
|
515 if (isempty (vartype)) |
|
5233
|
516 vartype = repmat ("C", nx, 1); |
|
|
517 elseif (! ischar (vartype) || all (size (vartype) > 1) |
|
|
518 || length (vartype) != nx) |
|
5244
|
519 error ("VARTYPE must be a char valued vector of length %d", nx); |
|
5233
|
520 return; |
|
|
521 elseif (! all (vartype == "C" | vartype == "I")) |
|
|
522 error ("VARTYPE must contain only C or I"); |
|
|
523 return; |
|
|
524 endif |
|
|
525 else |
|
|
526 ## As default we consider continuous vars |
|
|
527 vartype = repmat ("C", nx, 1); |
|
|
528 endif |
|
5232
|
529 |
|
5289
|
530 ## 8) Sense of optimization |
|
|
531 |
|
|
532 if (nargin > 7) |
|
|
533 if (isempty (sense)) |
|
|
534 sense = 1; |
|
|
535 elseif (ischar (sense) || all (size (sense) > 1) || ! isreal (sense)) |
|
|
536 error ("SENSE must be an integer value"); |
|
|
537 elseif (sense >= 0) |
|
|
538 sense = 1; |
|
|
539 else |
|
|
540 sense = -1; |
|
|
541 endif |
|
|
542 else |
|
|
543 sense = 1; |
|
|
544 endif |
|
|
545 |
|
|
546 ## 9) Parameters vector |
|
5233
|
547 |
|
|
548 if (nargin > 8) |
|
|
549 if (! isstruct (param)) |
|
|
550 error ("PARAM must be a structure"); |
|
|
551 return; |
|
|
552 endif |
|
|
553 else |
|
|
554 param = struct (); |
|
|
555 endif |
|
5232
|
556 |
|
5233
|
557 [xopt, fmin, status, extra] = ... |
|
5237
|
558 __glpk__ (c, a, b, lb, ub, ctype, vartype, sense, param); |
|
5232
|
559 |
|
|
560 endfunction |
