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1 ## Copyright (C) 2005 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 2, or (at your option) |
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8 ## 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, write to the Free |
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17 ## Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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18 ## 02110-1301, USA. |
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19 |
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20 ## -*- texinfo -*- |
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21 ## @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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22 ## Solve a linear program using the GNU GLPK library. Given three |
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23 ## arguments, @code{glpk} solves the following standard LP: |
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24 ## |
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25 ## @iftex |
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26 ## @tex |
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27 ## $$ |
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28 ## \min_x C^T x |
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29 ## $$ |
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30 ## @end tex |
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31 ## @end iftex |
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32 ## @ifnottex |
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33 ## @example |
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34 ## min C'*x |
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35 ## @end example |
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36 ## @end ifnottex |
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37 ## |
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38 ## subject to |
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39 ## |
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40 ## @iftex |
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41 ## @tex |
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42 ## $$ |
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43 ## Ax = b \qquad x \geq 0 |
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44 ## $$ |
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45 ## @end tex |
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46 ## @end iftex |
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47 ## @ifnottex |
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48 ## @example |
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49 ## @group |
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50 ## A*x = b |
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51 ## x >= 0 |
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52 ## @end group |
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53 ## @end example |
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54 ## @end ifnottex |
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55 ## |
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56 ## but may also solve problems of the form |
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57 ## |
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58 ## @iftex |
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59 ## @tex |
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60 ## $$ |
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61 ## [ \min_x | \max_x ] C^T x |
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62 ## $$ |
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63 ## @end tex |
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64 ## @end iftex |
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65 ## @ifnottex |
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66 ## @example |
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67 ## [ min | max ] C'*x |
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68 ## @end example |
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69 ## @end ifnottex |
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70 ## |
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71 ## subject to |
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72 ## |
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73 ## @iftex |
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74 ## @tex |
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75 ## $$ |
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76 ## Ax [ = | \leq | \geq ] b \qquad LB \leq x \leq UB |
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77 ## $$ |
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78 ## @end tex |
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79 ## @end iftex |
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80 ## @ifnottex |
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81 ## @example |
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82 ## @group |
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83 ## A*x [ "=" | "<=" | ">=" ] b |
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84 ## x >= LB |
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85 ## x <= UB |
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86 ## @end group |
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87 ## @end example |
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88 ## @end ifnottex |
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89 ## |
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90 ## Input arguments: |
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91 ## |
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92 ## @table @var |
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93 ## @item c |
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94 ## A column array containing the objective function coefficients. |
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95 ## |
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96 ## @item a |
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97 ## A matrix containing the constraints coefficients. |
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98 ## |
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99 ## @item b |
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100 ## A column array containing the right-hand side value for each constraint |
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101 ## in the constraint matrix. |
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102 ## |
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103 ## @item lb |
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104 ## An array containing the lower bound on each of the variables. If |
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105 ## @var{lb} is not supplied, the default lower bound for the variables is |
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106 ## zero. |
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107 ## |
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108 ## @item ub |
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109 ## An array containing the upper bound on each of the variables. If |
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110 ## @var{ub} is not supplied, the default upper bound is assumed to be |
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111 ## infinite. |
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112 ## |
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113 ## @item ctype |
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114 ## An array of characters containing the sense of each constraint in the |
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115 ## constraint matrix. Each element of the array may be one of the |
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116 ## following values |
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117 ## @table @code |
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118 ## @item "F" |
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119 ## Free (unbounded) variable (the constraint is ignored). |
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120 ## @item "U" |
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121 ## Variable with upper bound (@code{A(i,:)*x <= b(i)}). |
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122 ## @item "S" |
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123 ## Fixed Variable (@code{A(i,:)*x = b(i)}). |
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124 ## @item "L" |
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125 ## Variable with lower bound (@code{A(i,:)*x >= b(i)}). |
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126 ## @item "D" |
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127 ## Double-bounded variable (@code{A(i,:)*x >= -b(i)} @emph{and} |
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128 ## (@code{A(i,:)*x <= b(i)}). |
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129 ## @end table |
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130 ## |
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131 ## @item vartype |
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132 ## A column array containing the types of the variables. |
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133 ## @table @code |
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134 ## @item "F" |
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135 ## "C" |
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136 ## Continuous variable. |
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137 ## "I" |
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138 ## Integer variable |
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139 ## @end table |
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140 ## |
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141 ## @item sense |
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142 ## If @var{sense} is 1, the problem is a minimization. If @var{sense} is |
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143 ## -1, the problem is a maximization. The default value is 1. |
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144 ## |
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145 ## @item param |
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146 ## A structure containing the following parameters used to define the |
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147 ## behavior of solver. Missing elements in the structure take on default |
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148 ## values, so you only need to set the elements that you wish to change |
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149 ## from the default. |
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150 ## |
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151 ## Integer parameters: |
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152 ## |
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153 ## @table @code |
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154 ## @item msglev (@code{LPX_K_MSGLEV}, default: 1) |
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155 ## Level of messages output by solver routines: |
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156 ## @table @asis |
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157 ## @item 0 |
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158 ## No output. |
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159 ## @item 1 |
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160 ## Error messages only. |
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161 ## @item 2 |
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162 ## Normal output . |
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163 ## @item 3 |
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164 ## Full output (includes informational messages). |
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165 ## @end table |
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166 ## |
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167 ## @item scale (@code{LPX_K_SCALE}, default: 1) |
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168 ## Scaling option: |
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169 ## @table @asis |
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170 ## @item 0 |
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171 ## No scaling . |
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172 ## @item 1 |
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173 ## Equilibration scaling . |
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174 ## @item 2 |
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175 ## Geometric mean scaling, then equilibration scaling. |
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176 ## @end table |
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177 ## |
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178 ## @item dual (@code{LPX_K_DUAL}, default: 0) |
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179 ## Dual simplex option: |
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180 ## @table @asis |
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181 ## @item 0 |
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182 ## Do not use the dual simplex. |
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183 ## @item 1 |
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184 ## If initial basic solution is dual feasible, use the dual simplex. |
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185 ## @end table |
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186 ## |
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187 ## @item price (@code{LPX_K_PRICE}, default: 1) |
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188 ## Pricing option (for both primal and dual simplex): |
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189 ## @table @asis |
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190 ## @item 0 |
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191 ## Textbook pricing. |
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192 ## @item 1 |
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193 ## Steepest edge pricing. |
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194 ## @end table |
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195 ## |
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196 ## @item round (@code{LPX_K_ROUND}, default: 0) |
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197 ## Solution rounding option: |
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198 ## @table @asis |
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199 ## @item 0 |
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200 ## Report all primal and dual values "as is". |
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201 ## @item 1 |
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202 ## Replace tiny primal and dual values by exact zero. |
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203 ## @end table |
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204 ## |
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205 ## @item itlim (@code{LPX_K_ITLIM}, default: -1) |
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206 ## Simplex iterations limit. If this value is positive, it is decreased by |
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207 ## one each time when one simplex iteration has been performed, and |
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208 ## reaching zero value signals the solver to stop the search. Negative |
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209 ## value means no iterations limit. |
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210 ## |
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211 ## @item itcnt (@code{LPX_K_OUTFRQ}, default: 200) |
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212 ## Output frequency, in iterations. This parameter specifies how |
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213 ## frequently the solver sends information about the solution to the |
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214 ## standard output. |
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215 ## |
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216 ## @item branch (@code{LPX_K_BRANCH}, default: 2) |
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217 ## Branching heuristic option (for MIP only): |
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218 ## @table @asis |
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219 ## @item 0 |
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220 ## Branch on the first variable. |
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221 ## @item 1 |
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222 ## Branch on the last variable. |
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223 ## @item 2 |
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224 ## Branch using a heuristic by Driebeck and Tomlin. |
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225 ## @end table |
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226 ## |
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227 ## @item btrack (@code{LPX_K_BTRACK}, default: 2) |
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228 ## Backtracking heuristic option (for MIP only): |
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229 ## @table @asis |
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230 ## @item 0 |
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231 ## Depth first search. |
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232 ## @item 1 |
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233 ## Breadth first search. |
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234 ## @item 2 |
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235 ## Backtrack using the best projection heuristic. |
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236 ## @end table |
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237 ## |
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238 ## @item presol (@code{LPX_K_PRESOL}, default: 1) |
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239 ## If this flag is set, the routine lpx_simplex solves the problem using |
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240 ## the built-in LP presolver. Otherwise the LP presolver is not used. |
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241 ## |
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242 ## @item lpsolver (default: 1) |
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243 ## Select which solver to use. If the problem is a MIP problem this flag |
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244 ## will be ignored. |
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245 ## @table @asis |
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246 ## @item 1 |
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247 ## Revised simplex method. |
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248 ## @item 2 |
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249 ## Interior point method. |
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250 ## @end table |
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251 ## @item save (default: 0) |
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252 ## If this parameter is nonzero, save a copy of the problem problem in |
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253 ## CPLEX LP format to the file @file{"outpb.lp"}. There is currently no |
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254 ## way to change the name of the output file. |
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255 ## @end table |
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256 ## |
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257 ## Real parameters: |
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258 ## |
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259 ## @table @code |
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260 ## @item relax (@code{LPX_K_RELAX}, default: 0.07) |
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261 ## Relaxation parameter used in the ratio test. If it is zero, the textbook |
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262 ## ratio test is used. If it is non-zero (should be positive), Harris' |
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263 ## two-pass ratio test is used. In the latter case on the first pass of the |
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264 ## ratio test basic variables (in the case of primal simplex) or reduced |
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265 ## costs of non-basic variables (in the case of dual simplex) are allowed |
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266 ## to slightly violate their bounds, but not more than |
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267 ## @code{relax*tolbnd} or @code{relax*toldj (thus, @code{relax} is a |
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268 ## percentage of @code{tolbnd} or @code{toldj}}. |
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269 ## |
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270 ## @item tolbnd (@code{LPX_K_TOLBND}, default: 10e-7) |
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271 ## Relative tolerance used to check if the current basic solution is primal |
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272 ## feasible. It is not recommended that you change this parameter unless you |
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273 ## have a detailed understanding of its purpose. |
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274 ## |
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275 ## @item toldj (@code{LPX_K_TOLDJ}, default: 10e-7) |
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276 ## Absolute tolerance used to check if the current basic solution is dual |
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277 ## feasible. It is not recommended that you change this parameter unless you |
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278 ## have a detailed understanding of its purpose. |
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279 ## |
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280 ## @item tolpiv (@code{LPX_K_TOLPIV}, default: 10e-9) |
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281 ## Relative tolerance used to choose eligible pivotal elements of the |
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282 ## simplex table. It is not recommended that you change this parameter unless you |
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283 ## have a detailed understanding of its purpose. |
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284 ## |
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285 ## @item objll (@code{LPX_K_OBJLL}, default: -DBL_MAX) |
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286 ## Lower limit of the objective function. If on the phase II the objective |
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287 ## function reaches this limit and continues decreasing, the solver stops |
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288 ## the search. This parameter is used in the dual simplex method only. |
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289 ## |
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290 ## @item objul (@code{LPX_K_OBJUL}, default: +DBL_MAX) |
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291 ## Upper limit of the objective function. If on the phase II the objective |
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292 ## function reaches this limit and continues increasing, the solver stops |
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293 ## the search. This parameter is used in the dual simplex only. |
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294 ## |
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295 ## @item tmlim (@code{LPX_K_TMLIM}, default: -1.0) |
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296 ## Searching time limit, in seconds. If this value is positive, it is |
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297 ## decreased each time when one simplex iteration has been performed by the |
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298 ## amount of time spent for the iteration, and reaching zero value signals |
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299 ## the solver to stop the search. Negative value means no time limit. |
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300 ## |
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301 ## @item outdly (@code{LPX_K_OUTDLY}, default: 0.0) |
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302 ## Output delay, in seconds. This parameter specifies how long the solver |
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303 ## should delay sending information about the solution to the standard |
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304 ## output. Non-positive value means no delay. |
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305 ## |
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306 ## @item tolint (@code{LPX_K_TOLINT}, default: 10e-5) |
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307 ## Relative tolerance used to check if the current basic solution is integer |
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308 ## feasible. It is not recommended that you change this parameter unless |
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309 ## you have a detailed understanding of its purpose. |
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310 ## |
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311 ## @item tolobj (@code{LPX_K_TOLOBJ}, default: 10e-7) |
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312 ## Relative tolerance used to check if the value of the objective function |
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313 ## is not better than in the best known integer feasible solution. It is |
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314 ## not recommended that you change this parameter unless you have a |
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315 ## detailed understanding of its purpose. |
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316 ## @end table |
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317 ## @end table |
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318 ## |
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319 ## Output values: |
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320 ## |
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321 ## @table @var |
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322 ## @item xopt |
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323 ## The optimizer (the value of the decision variables at the optimum). |
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324 ## @item fopt |
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325 ## The optimum value of the objective function. |
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326 ## @item status |
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327 ## Status of the optimization. |
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328 ## |
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329 ## Simplex Method: |
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330 ## @table @asis |
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331 ## @item 180 (@code{LPX_OPT}) |
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332 ## Solution is optimal. |
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333 ## @item 181 (@code{LPX_FEAS}) |
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334 ## Solution is feasible. |
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335 ## @item 182 (@code{LPX_INFEAS}) |
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336 ## Solution is infeasible. |
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337 ## @item 183 (@code{LPX_NOFEAS}) |
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338 ## Problem has no feasible solution. |
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339 ## @item 184 (@code{LPX_UNBND}) |
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340 ## Problem has no unbounded solution. |
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341 ## @item 185 (@code{LPX_UNDEF}) |
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342 ## Solution status is undefined. |
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343 ## @end table |
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344 ## Interior Point Method: |
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345 ## @table @asis |
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346 ## @item 150 (@code{LPX_T_UNDEF}) |
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347 ## The interior point method is undefined. |
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348 ## @item 151 (@code{LPX_T_OPT}) |
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349 ## The interior point method is optimal. |
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350 ## @end table |
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351 ## Mixed Integer Method: |
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352 ## @table @asis |
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353 ## @item 170 (@code{LPX_I_UNDEF}) |
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354 ## The status is undefined. |
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355 ## @item 171 (@code{LPX_I_OPT}) |
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356 ## The solution is integer optimal. |
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357 ## @item 172 (@code{LPX_I_FEAS}) |
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358 ## Solution integer feasible but its optimality has not been proven |
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359 ## @item 173 (@code{LPX_I_NOFEAS}) |
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360 ## No integer feasible solution. |
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361 ## @end table |
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362 ## @noindent |
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363 ## If an error occurs, @var{status} will contain one of the following |
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364 ## codes: |
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365 ## |
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366 ## @table @asis |
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367 ## @item 204 (@code{LPX_E_FAULT}) |
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368 ## Unable to start the search. |
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369 ## @item 205 (@code{LPX_E_OBJLL}) |
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370 ## Objective function lower limit reached. |
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371 ## @item 206 (@code{LPX_E_OBJUL}) |
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372 ## Objective function upper limit reached. |
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373 ## @item 207 (@code{LPX_E_ITLIM}) |
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374 ## Iterations limit exhausted. |
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375 ## @item 208 (@code{LPX_E_TMLIM}) |
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376 ## Time limit exhausted. |
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377 ## @item 209 (@code{LPX_E_NOFEAS}) |
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378 ## No feasible solution. |
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379 ## @item 210 (@code{LPX_E_INSTAB}) |
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380 ## Numerical instability. |
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381 ## @item 211 (@code{LPX_E_SING}) |
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382 ## Problems with basis matrix. |
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383 ## @item 212 (@code{LPX_E_NOCONV}) |
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384 ## No convergence (interior). |
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385 ## @item 213 (@code{LPX_E_NOPFS}) |
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386 ## No primal feasible solution (LP presolver). |
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387 ## @item 214 (@code{LPX_E_NODFS}) |
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388 ## No dual feasible solution (LP presolver). |
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389 ## @end table |
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390 ## @item extra |
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391 ## A data structure containing the following fields: |
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392 ## @table @code |
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393 ## @item lambda |
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394 ## Dual variables. |
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395 ## @item redcosts |
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396 ## Reduced Costs. |
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397 ## @item time |
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398 ## Time (in seconds) used for solving LP/MIP problem. |
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399 ## @item mem |
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400 ## Memory (in bytes) used for solving LP/MIP problem (this is not |
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401 ## available if the version of GLPK is 4.15 or later). |
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402 ## @end table |
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403 ## @end table |
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404 ## |
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405 ## Example: |
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406 ## |
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407 ## @example |
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408 ## @group |
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409 ## c = [10, 6, 4]'; |
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410 ## a = [ 1, 1, 1; |
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411 ## 10, 4, 5; |
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412 ## 2, 2, 6]; |
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413 ## b = [100, 600, 300]'; |
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414 ## lb = [0, 0, 0]'; |
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415 ## ub = []; |
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416 ## ctype = "UUU"; |
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417 ## vartype = "CCC"; |
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418 ## s = -1; |
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419 ## |
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420 ## param.msglev = 1; |
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421 ## param.itlim = 100; |
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422 ## |
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423 ## [xmin, fmin, status, extra] = ... |
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424 ## glpk (c, a, b, lb, ub, ctype, vartype, s, param); |
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425 ## @end group |
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426 ## @end example |
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427 ## @end deftypefn |
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428 |
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429 ## Author: Nicolo' Giorgetti <giorgetti@dii.unisi.it> |
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430 ## Adapted-by: jwe |
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431 |
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432 function [xopt, fmin, status, extra] = glpk (c, a, b, lb, ub, ctype, vartype, sense, param) |
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433 |
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434 ## If there is no input output the version and syntax |
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435 if (nargin < 3 || nargin > 9) |
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436 print_usage (); |
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5233
|
437 return; |
|
|
438 endif |
|
5232
|
439 |
|
5233
|
440 if (all (size (c) > 1) || iscomplex (c) || ischar (c)) |
|
|
441 error ("C must be a real vector"); |
|
|
442 return; |
|
|
443 endif |
|
|
444 nx = length (c); |
|
|
445 ## Force column vector. |
|
|
446 c = c(:); |
|
5232
|
447 |
|
5237
|
448 ## 2) Matrix constraint |
|
5232
|
449 |
|
5233
|
450 if (isempty (a)) |
|
|
451 error ("A cannot be an empty matrix"); |
|
|
452 return; |
|
|
453 endif |
|
|
454 [nc, nxa] = size(a); |
|
|
455 if (! isreal (a) || nxa != nx) |
|
|
456 error ("A must be a real valued %d by %d matrix", nc, nx); |
|
|
457 return; |
|
|
458 endif |
|
|
459 |
|
5237
|
460 ## 3) RHS |
|
5232
|
461 |
|
5233
|
462 if (isempty (b)) |
|
|
463 error ("B cannot be an empty vector"); |
|
|
464 return; |
|
|
465 endif |
|
|
466 if (! isreal (b) || length (b) != nc) |
|
|
467 error ("B must be a real valued %d by 1 vector", nc); |
|
|
468 return; |
|
|
469 endif |
|
|
470 |
|
5237
|
471 ## 4) Vector with the lower bound of each variable |
|
5232
|
472 |
|
5237
|
473 if (nargin > 3) |
|
5233
|
474 if (isempty (lb)) |
|
5237
|
475 lb = zeros (0, nx, 1); |
|
5233
|
476 elseif (! isreal (lb) || all (size (lb) > 1) || length (lb) != nx) |
|
|
477 error ("LB must be a real valued %d by 1 column vector", nx); |
|
|
478 return; |
|
|
479 endif |
|
|
480 else |
|
5237
|
481 lb = zeros (nx, 1); |
|
5233
|
482 end |
|
|
483 |
|
5237
|
484 ## 5) Vector with the upper bound of each variable |
|
5232
|
485 |
|
5237
|
486 if (nargin > 4) |
|
5233
|
487 if (isempty (ub)) |
|
|
488 ub = repmat (Inf, nx, 1); |
|
|
489 elseif (! isreal (ub) || all (size (ub) > 1) || length (ub) != nx) |
|
|
490 error ("UB must be a real valued %d by 1 column vector", nx); |
|
|
491 return; |
|
|
492 endif |
|
|
493 else |
|
|
494 ub = repmat (Inf, nx, 1); |
|
|
495 end |
|
5232
|
496 |
|
5237
|
497 ## 6) Sense of each constraint |
|
5232
|
498 |
|
5237
|
499 if (nargin > 5) |
|
|
500 if (isempty (ctype)) |
|
|
501 ctype = repmat ("S", nc, 1); |
|
|
502 elseif (! ischar (ctype) || all (size (ctype) > 1) || length (ctype) != nc) |
|
5244
|
503 error ("CTYPE must be a char valued vector of length %d", nc); |
|
5237
|
504 return; |
|
|
505 elseif (! all (ctype == "F" | ctype == "U" | ctype == "S" |
|
|
506 | ctype == "L" | ctype == "D")) |
|
|
507 error ("CTYPE must contain only F, U, S, L, or D"); |
|
|
508 return; |
|
|
509 endif |
|
|
510 else |
|
|
511 ctype = repmat ("S", nc, 1); |
|
|
512 end |
|
|
513 |
|
|
514 ## 7) Vector with the type of variables |
|
|
515 |
|
|
516 if (nargin > 6) |
|
5289
|
517 if (isempty (vartype)) |
|
5233
|
518 vartype = repmat ("C", nx, 1); |
|
|
519 elseif (! ischar (vartype) || all (size (vartype) > 1) |
|
|
520 || length (vartype) != nx) |
|
5244
|
521 error ("VARTYPE must be a char valued vector of length %d", nx); |
|
5233
|
522 return; |
|
|
523 elseif (! all (vartype == "C" | vartype == "I")) |
|
|
524 error ("VARTYPE must contain only C or I"); |
|
|
525 return; |
|
|
526 endif |
|
|
527 else |
|
|
528 ## As default we consider continuous vars |
|
|
529 vartype = repmat ("C", nx, 1); |
|
|
530 endif |
|
5232
|
531 |
|
5289
|
532 ## 8) Sense of optimization |
|
|
533 |
|
|
534 if (nargin > 7) |
|
|
535 if (isempty (sense)) |
|
|
536 sense = 1; |
|
|
537 elseif (ischar (sense) || all (size (sense) > 1) || ! isreal (sense)) |
|
|
538 error ("SENSE must be an integer value"); |
|
|
539 elseif (sense >= 0) |
|
|
540 sense = 1; |
|
|
541 else |
|
|
542 sense = -1; |
|
|
543 endif |
|
|
544 else |
|
|
545 sense = 1; |
|
|
546 endif |
|
|
547 |
|
|
548 ## 9) Parameters vector |
|
5233
|
549 |
|
|
550 if (nargin > 8) |
|
|
551 if (! isstruct (param)) |
|
|
552 error ("PARAM must be a structure"); |
|
|
553 return; |
|
|
554 endif |
|
|
555 else |
|
|
556 param = struct (); |
|
|
557 endif |
|
5232
|
558 |
|
5233
|
559 [xopt, fmin, status, extra] = ... |
|
5237
|
560 __glpk__ (c, a, b, lb, ub, ctype, vartype, sense, param); |
|
5232
|
561 |
|
|
562 endfunction |
