1 | %* glpk05.tex *% |
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2 | |
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3 | \chapter{Branch-and-Cut API Routines} |
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4 | |
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5 | \section{Introduction} |
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6 | |
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7 | \subsection{Using the callback routine} |
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8 | |
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9 | The GLPK MIP solver based on the branch-and-cut method allows the |
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10 | application program to control the solution process. This is attained |
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11 | by means of the user-defined callback routine, which is called by the |
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12 | solver at various points of the branch-and-cut algorithm. |
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13 | |
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14 | The callback routine passed to the MIP solver should be written by the |
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15 | user and has the following specification:\footnote{The name |
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16 | {\tt foo\_bar} used here is a placeholder for the callback routine |
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17 | name.} |
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18 | |
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19 | \begin{verbatim} |
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20 | void foo_bar(glp_tree *tree, void *info); |
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21 | \end{verbatim} |
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22 | |
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23 | \noindent |
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24 | where \verb|tree| is a pointer to the data structure \verb|glp_tree|, |
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25 | which should be used on subsequent calls to branch-and-cut interface |
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26 | routines, and \verb|info| is a transit pointer passed to the routine |
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27 | \verb|glp_intopt|, which may be used by the application program to pass |
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28 | some external data to the callback routine. |
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29 | |
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30 | The callback routine is passed to the MIP solver through the control |
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31 | parameter structure \verb|glp_iocp| (see Chapter ``Basic API Routines'', |
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32 | Section ``Mixed integer programming routines'', Subsection ``Solve MIP |
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33 | problem with the branch-and-cut method'') as follows: |
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34 | |
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35 | \newpage |
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36 | |
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37 | \begin{verbatim} |
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38 | glp_prob *mip; |
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39 | glp_iocp parm; |
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40 | . . . |
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41 | glp_init_iocp(&parm); |
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42 | . . . |
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43 | parm.cb_func = foo_bar; |
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44 | parm.cb_info = ... ; |
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45 | ret = glp_intopt(mip, &parm); |
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46 | . . . |
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47 | \end{verbatim} |
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48 | |
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49 | To determine why it is being called by the MIP solver the callback |
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50 | routine should use the routine \verb|glp_ios_reason| (described in this |
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51 | section below), which returns a code indicating the reason for calling. |
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52 | Depending on the reason the callback routine may perform necessary |
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53 | actions to control the solution process. |
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54 | |
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55 | The reason codes, which correspond to various point of the |
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56 | branch-and-cut algorithm implemented in the MIP solver, are described |
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57 | in Subsection ``Reasons for calling the callback routine'' below. |
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58 | |
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59 | To ignore calls for reasons, which are not processed by the callback |
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60 | routine, it should just return to the MIP solver doing nothing. For |
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61 | example: |
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62 | |
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63 | \begin{verbatim} |
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64 | void foo_bar(glp_tree *tree, void *info) |
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65 | { . . . |
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66 | switch (glp_ios_reason(tree)) |
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67 | { case GLP_IBRANCH: |
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68 | . . . |
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69 | break; |
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70 | case GLP_ISELECT: |
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71 | . . . |
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72 | break; |
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73 | default: |
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74 | /* ignore call for other reasons */ |
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75 | break; |
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76 | } |
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77 | return; |
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78 | } |
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79 | \end{verbatim} |
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80 | |
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81 | To control the solution process as well as to obtain necessary |
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82 | information the callback routine may use the branch-and-cut API |
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83 | routines described in this chapter. Names of all these routines begin |
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84 | with `\verb|glp_ios_|'. |
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85 | |
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86 | \subsection{Branch-and-cut algorithm} |
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87 | |
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88 | This section gives a schematic description of the branch-and-cut |
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89 | algorithm as it is implemented in the GLPK MIP solver. |
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90 | |
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91 | \medskip |
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92 | |
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93 | {\it 1. Initialization} |
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94 | |
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95 | Set $L:=\{P_0\}$, where $L$ is the {\it active list} (i.e. the list of |
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96 | active subproblems), $P_0$ is the original MIP problem to be solved. |
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97 | |
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98 | Set $z^{\it best}:=+\infty$ (in case of minimization) or |
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99 | $z^{\it best}:=-\infty$ (in case of maximization), where $z^{\it best}$ |
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100 | is {\it incumbent value}, i.e. an upper (minimization) or lower |
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101 | (maximization) global bound for $z^{\it opt}$, the optimal objective |
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102 | value for $P^0$. |
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103 | |
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104 | \medskip |
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105 | |
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106 | {\it 2. Subproblem selection} |
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107 | |
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108 | If $L=\varnothing$ then GO TO 9. |
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109 | |
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110 | Select $P\in L$, i.e. make active subproblem $P$ current. |
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111 | |
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112 | \medskip |
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113 | |
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114 | {\it 3. Solving LP relaxation} |
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115 | |
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116 | Solve $P^{\it LP}$, which is LP relaxation of $P$. |
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117 | |
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118 | If $P^{\it LP}$ has no primal feasible solution then GO TO 8. |
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119 | |
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120 | Let $z^{\it LP}$ be the optimal objective value for $P^{\it LP}$. |
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121 | |
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122 | If $z^{\it LP}\geq z^{\it best}$ (in case of minimization) or |
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123 | $z^{\it LP}\leq z^{\rm best}$ (in case of maximization) then GO TO 8. |
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124 | |
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125 | \medskip |
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126 | |
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127 | {\it 4. Adding ``lazy'' constraints} |
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128 | |
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129 | Let $x^{\it LP}$ be the optimal solution to $P^{\it LP}$. |
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130 | |
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131 | If there are ``lazy'' constraints (i.e. essential constraints not |
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132 | included in the original MIP problem $P_0$), which are violated at the |
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133 | optimal point $x^{\it LP}$, add them to $P$, and GO TO 3. |
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134 | |
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135 | \medskip |
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136 | |
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137 | {\it 5. Check for integrality} |
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138 | |
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139 | Let $x_j$ be a variable, which is required to be integer, and let |
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140 | $x^{\it LP}_j\in x^{\it LP}$ be its value in the optimal solution to |
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141 | $P^{\it LP}$. |
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142 | |
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143 | If $x^{\it LP}_j$ are integral for all integer variables, then a better |
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144 | integer feasible solution is found. Store its components, set |
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145 | $z^{\it best}:=z^{\it LP}$, and GO TO 8. |
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146 | |
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147 | \medskip |
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148 | |
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149 | {\it 6. Adding cutting planes} |
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150 | |
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151 | If there are cutting planes (i.e. valid constraints for $P$), |
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152 | which are violated at the optimal point $x^{\it LP}$, add them to $P$, |
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153 | and GO TO 3. |
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154 | |
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155 | \medskip |
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156 | |
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157 | {\it 7. Branching} |
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158 | |
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159 | Select {\it branching variable} $x_j$, i.e. a variable, which is |
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160 | required to be integer, and whose value $x^{\it LP}_j\in x^{\it LP}$ is |
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161 | fractional in the optimal solution to $P^{\it LP}$. |
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162 | |
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163 | Create new subproblem $P^D$ (so called {\it down branch}), which is |
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164 | identical to the current subproblem $P$ with exception that the upper |
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165 | bound of $x_j$ is replaced by $\lfloor x^{\it LP}_j\rfloor$. (For |
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166 | example, if $x^{\it LP}_j=3.14$, the new upper bound of $x_j$ in the |
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167 | down branch will be $\lfloor 3.14\rfloor=3$.) |
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168 | |
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169 | Create new subproblem $P^U$ (so called {\it up branch}), which is |
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170 | identical to the current subproblem $P$ with exception that the lower |
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171 | bound of $x_j$ is replaced by $\lceil x^{\it LP}_j\rceil$. (For example, |
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172 | if $x^{\it LP}_j=3.14$, the new lower bound of $x_j$ in the up branch |
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173 | will be $\lceil 3.14\rceil=4$.) |
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174 | |
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175 | Set $L:=(L\backslash\{P\})\cup\{P^D,P^U\}$, i.e. remove the current |
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176 | subproblem $P$ from the active list $L$ and add two new subproblems |
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177 | $P^D$ and $P^U$ to it. Then GO TO 2. |
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178 | |
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179 | \medskip |
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180 | |
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181 | {\it 8. Pruning} |
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182 | |
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183 | Remove from the active list $L$ all subproblems (including the current |
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184 | one), whose local bound $\widetilde{z}$ is not better than the global |
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185 | bound $z^{\it best}$, i.e. set $L:=L\backslash\{P\}$ for all $P$, where |
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186 | $\widetilde{z}\geq z^{\it best}$ (in case of minimization) or |
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187 | $\widetilde{z}\leq z^{\it best}$ (in case of maximization), and then |
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188 | GO TO 2. |
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189 | |
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190 | The local bound $\widetilde{z}$ for subproblem $P$ is an lower |
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191 | (minimization) or upper (maximization) bound for integer optimal |
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192 | solution to {\it this} subproblem (not to the original problem). This |
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193 | bound is local in the sense that only subproblems in the subtree rooted |
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194 | at node $P$ cannot have better integer feasible solutions. Note that |
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195 | the local bound is not necessarily the optimal objective value to LP |
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196 | relaxation $P^{\it LP}$. |
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197 | |
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198 | \medskip |
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199 | |
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200 | {\it 9. Termination} |
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201 | |
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202 | If $z^{\it best}=+\infty$ (in case of minimization) or |
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203 | $z^{\it best}=-\infty$ (in case of maximization), the original problem |
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204 | $P_0$ has no integer feasible solution. Otherwise, the last integer |
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205 | feasible solution stored on step 5 is the integer optimal solution to |
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206 | the original problem $P_0$ with $z^{\it opt}=z^{\it best}$. STOP. |
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207 | |
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208 | \subsection{The search tree} |
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209 | |
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210 | On the branching step of the branch-and-cut algorithm the current |
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211 | subproblem is divided into two\footnote{In more general cases the |
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212 | current subproblem may be divided into more than two subproblems. |
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213 | However, currently such feature is not used in GLPK.} new subproblems, |
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214 | so the set of all subproblems can be represented in the form of a rooted |
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215 | tree, which is called the {\it search} or {\it branch-and-bound} tree. |
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216 | An example of the search tree is shown on Fig.~1. Each node of the |
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217 | search tree corresponds to a subproblem, so the terms `node' and |
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218 | `subproblem' may be used synonymously. |
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219 | |
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220 | \newpage |
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221 | |
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222 | \begin{figure}[t] |
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223 | \noindent\hfil |
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224 | \xymatrix @R=20pt @C=10pt |
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225 | {&&&&&&*+<14pt>[o][F=]{A}\ar@{-}[dllll]\ar@{-}[dr]\ar@{-}[drrrr]&&&&\\ |
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226 | &&*+<14pt>[o][F=]{B}\ar@{-}[dl]\ar@{-}[dr]&&&&&*+<14pt>[o][F=]{C} |
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227 | \ar@{-}[dll]\ar@{-}[dr]\ar@{-}[drrr]&&&*+<14pt>[o][F-]{\times}\\ |
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228 | &*+<14pt>[o][F-]{\times}\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&& |
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229 | *+<14pt>[o][F-]{D}&&*+<14pt>[o][F=]{E}\ar@{-}[dl]\ar@{-}[dr]&&& |
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230 | *+<14pt>[o][F=]{F}\ar@{-}[dl]\ar@{-}[dr]&&*+<14pt>[o][F-]{G}\\ |
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231 | *+<14pt>[o][F-]{\times}&*+<14pt>[o][F-]{\times}&*+<14pt>[o][F-]{\times} |
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232 | &&*+<14pt>[][F-]{H}&&*+<14pt>[o][F-]{I}&*+<14pt>[o][F-]{\times}&& |
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233 | *+<14pt>[o][F-]{J}&\\} |
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234 | |
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235 | \bigskip |
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236 | |
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237 | \noindent\hspace{.8in} |
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238 | \xymatrix @R=11pt |
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239 | {*+<20pt>[][F-]{}&*\txt{\makebox[1in][l]{Current}}&& |
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240 | *+<20pt>[o][F-]{}&*\txt{\makebox[1in][l]{Active}}\\ |
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241 | *+<20pt>[o][F=]{}&*\txt{\makebox[1in][l]{Non-active}}&& |
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242 | *+<14pt>[o][F-]{\times}&*\txt{\makebox[1in][l]{Fathomed}}\\ |
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243 | } |
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244 | |
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245 | \begin{center} |
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246 | Fig. 1. An example of the search tree. |
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247 | \end{center} |
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248 | \end{figure} |
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249 | |
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250 | In GLPK each node may have one of the following four statuses: |
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251 | |
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252 | $\bullet$ {\it current node} is the active node currently being |
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253 | processed; |
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254 | |
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255 | $\bullet$ {\it active node} is a leaf node, which still has to be |
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256 | processed; |
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257 | |
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258 | $\bullet$ {\it non-active node} is a node, which has been processed, |
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259 | but not fathomed; |
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260 | |
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261 | $\bullet$ {\it fathomed node} is a node, which has been processed and |
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262 | fathomed. |
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263 | |
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264 | In the data structure representing the search tree GLPK keeps only |
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265 | current, active, and non-active nodes. Once a node has been fathomed, |
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266 | it is removed from the tree data structure. |
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267 | |
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268 | Being created each node of the search tree is assigned a distinct |
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269 | positive integer called the {\it subproblem reference number}, which |
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270 | may be used by the application program to specify a particular node of |
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271 | the tree. The root node corresponding to the original problem to be |
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272 | solved is always assigned the reference number 1. |
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273 | |
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274 | \subsection{Current subproblem} |
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275 | |
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276 | The current subproblem is a MIP problem corresponding to the current |
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277 | node of the search tree. It is represented as the GLPK problem object |
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278 | (\verb|glp_prob|) that allows the application program using API routines |
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279 | to access its content in the standard way. If the MIP presolver is not |
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280 | used, it is the original problem object passed to the routine |
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281 | \verb|glp_intopt|; otherwise, it is an internal problem object built by |
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282 | the MIP presolver. |
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283 | |
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284 | Note that the problem object is used by the MIP solver itself during |
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285 | the solution process for various purposes (to solve LP relaxations, to |
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286 | perfom branching, etc.), and even if the MIP presolver is not used, the |
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287 | current content of the problem object may differ from its original |
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288 | content. For example, it may have additional rows, bounds of some rows |
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289 | and columns may be changed, etc. In particular, LP segment of the |
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290 | problem object corresponds to LP relaxation of the current subproblem. |
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291 | However, on exit from the MIP solver the content of the problem object |
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292 | is restored to its original state. |
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293 | |
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294 | To obtain information from the problem object the application program |
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295 | may use any API routines, which do not change the object. Using API |
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296 | routines, which change the problem object, is restricted to stipulated |
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297 | cases. |
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298 | |
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299 | \subsection{The cut pool} |
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300 | |
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301 | The {\it cut pool} is a set of cutting plane constraints maintained by |
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302 | the MIP solver. It is used by the GLPK cut generation routines and may |
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303 | be used by the application program in the same way, i.e. rather than |
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304 | to add cutting plane constraints directly to the problem object the |
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305 | application program may store them to the cut pool. In the latter case |
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306 | the solver looks through the cut pool, selects efficient constraints, |
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307 | and adds them to the problem object. |
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308 | |
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309 | \subsection{Reasons for calling the callback routine} |
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310 | |
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311 | The callback routine may be called by the MIP solver for the following |
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312 | reasons. |
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313 | |
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314 | \subsubsection*{Request for subproblem selection} |
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315 | |
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316 | The callback routine is called with the reason code \verb|GLP_ISELECT| |
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317 | if the current subproblem has been fathomed and therefore there is no |
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318 | current subproblem. |
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319 | |
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320 | In response the callback routine may select some subproblem from the |
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321 | active list and pass its reference number to the solver using the |
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322 | routine \verb|glp_ios_select_node|, in which case the solver continues |
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323 | the search from the specified active subproblem. If no selection is made |
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324 | by the callback routine, the solver uses a backtracking technique |
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325 | specified by the control parameter \verb|bt_tech|. |
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326 | |
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327 | To explore the active list (i.e. active nodes of the branch-and-bound |
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328 | tree) the callback routine may use the routines \verb|glp_ios_next_node| |
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329 | and \verb|glp_ios_prev_node|. |
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330 | |
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331 | \subsubsection*{Request for preprocessing} |
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332 | |
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333 | The callback routine is called with the reason code \verb|GLP_IPREPRO| |
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334 | if the current subproblem has just been selected from the active list |
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335 | and its LP relaxation is not solved yet. |
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336 | |
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337 | In response the callback routine may perform some preprocessing of the |
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338 | current subproblem like tightening bounds of some variables or removing |
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339 | bounds of some redundant constraints. |
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340 | |
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341 | \subsubsection*{Request for row generation} |
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342 | |
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343 | The callback routine is called with the reason code \verb|GLP_IROWGEN| |
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344 | if LP relaxation of the current subproblem has just been solved to |
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345 | optimality and its objective value is better than the best known integer |
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346 | feasible solution. |
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347 | |
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348 | In response the callback routine may add one or more ``lazy'' |
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349 | constraints (rows), which are violated by the current optimal solution |
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350 | of LP relaxation, using API routines \verb|glp_add_rows|, |
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351 | \verb|glp_set_row_name|, \verb|glp_set_row_bnds|, and |
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352 | \verb|glp_set_mat_row|, in which case the solver will perform |
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353 | re-optimization of LP relaxation. If there are no violated constraints, |
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354 | the callback routine should just return. |
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355 | |
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356 | Optimal solution components for LP relaxation can be obtained with API |
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357 | routines \verb|glp_get_obj_val|, \verb|glp_get_row_prim|, |
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358 | \verb|glp_get_row_dual|, \verb|glp_get_col_prim|, and |
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359 | \verb|glp_get_col_dual|. |
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360 | |
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361 | \subsubsection*{Request for heuristic solution} |
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362 | |
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363 | The callback routine is called with the reason code \verb|GLP_IHEUR| |
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364 | if LP relaxation of the current subproblem being solved to optimality |
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365 | is integer infeasible (i.e. values of some structural variables of |
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366 | integer kind are fractional), though its objective value is better than |
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367 | the best known integer feasible solution. |
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368 | |
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369 | In response the callback routine may try applying a primal heuristic |
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370 | to find an integer feasible solution,\footnote{Integer feasible to the |
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371 | original MIP problem, not to the current subproblem.} which is better |
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372 | than the best known one. In case of success the callback routine may |
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373 | store such better solution in the problem object using the routine |
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374 | \verb|glp_ios_heur_sol|. |
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375 | |
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376 | \subsubsection*{Request for cut generation} |
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377 | |
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378 | The callback routine is called with the reason code \verb|GLP_ICUTGEN| |
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379 | if LP relaxation of the current subproblem being solved to optimality |
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380 | is integer infeasible (i.e. values of some structural variables of |
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381 | integer kind are fractional), though its objective value is better than |
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382 | the best known integer feasible solution. |
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383 | |
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384 | In response the callback routine may reformulate the {\it current} |
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385 | subproblem (before it will be splitted up due to branching) by adding to |
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386 | the problem object one or more {\it cutting plane constraints}, which |
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387 | cut off the fractional optimal point from the MIP |
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388 | polytope.\footnote{Since these constraints are added to the current |
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389 | subproblem, they may be globally as well as locally valid.} |
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390 | |
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391 | Adding cutting plane constraints may be performed in two ways. |
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392 | One way is the same as for the reason code \verb|GLP_IROWGEN| (see |
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393 | above), in which case the callback routine adds new rows corresponding |
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394 | to cutting plane constraints directly to the current subproblem. |
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395 | |
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396 | The other way is to add cutting plane constraints to the {\it cut pool}, |
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397 | a set of cutting plane constraints maintained by the solver, rather than |
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398 | directly to the current subproblem. In this case after return from the |
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399 | callback routine the solver looks through the cut pool, selects |
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400 | efficient cutting plane constraints, adds them to the current |
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401 | subproblem, drops other constraints, and then performs re-optimization. |
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402 | |
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403 | \subsubsection*{Request for branching} |
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404 | |
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405 | The callback routine is called with the reason code \verb|GLP_IBRANCH| |
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406 | if LP relaxation of the current subproblem being solved to optimality |
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407 | is integer infeasible (i.e. values of some structural variables of |
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408 | integer kind are fractional), though its objective value is better than |
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409 | the best known integer feasible solution. |
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410 | |
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411 | In response the callback routine may choose some variable suitable for |
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412 | branching (i.e. integer variable, whose value in optimal solution to |
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413 | LP relaxation of the current subproblem is fractional) and pass its |
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414 | ordinal number to the solver using the routine |
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415 | \verb|glp_ios_branch_upon|, in which case the solver splits the current |
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416 | subproblem in two new subproblems and continues the search. If no choice |
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417 | is made by the callback routine, the solver uses a branching technique |
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418 | specified by the control parameter \verb|br_tech|. |
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419 | |
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420 | \subsubsection*{Better integer solution found} |
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421 | |
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422 | The callback routine is called with the reason code \verb|GLP_IBINGO| |
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423 | if LP relaxation of the current subproblem being solved to optimality |
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424 | is integer feasible (i.e. values of all structural variables of integer |
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425 | kind are integral within the working precision) and its objective value |
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426 | is better than the best known integer feasible solution. |
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427 | |
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428 | Optimal solution components for LP relaxation can be obtained in the |
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429 | same way as for the reason code \verb|GLP_IROWGEN| (see above). |
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430 | |
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431 | Components of the new MIP solution can be obtained with API routines |
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432 | \verb|glp_mip_obj_val|, \verb|glp_mip_row_val|, and |
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433 | \verb|glp_mip_col_val|. Note, however, that due to row/cut generation |
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434 | there may be additional rows in the problem object. |
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435 | |
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436 | The difference between optimal solution to LP relaxation and |
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437 | corresponding MIP solution is that in the former case some structural |
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438 | variables of integer kind (namely, basic variables) may have values, |
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439 | which are close to nearest integers within the working precision, while |
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440 | in the latter case all such variables have exact integral values. |
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441 | |
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442 | The reason \verb|GLP_IBINGO| is intended only for informational |
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443 | purposes, so the callback routine should not modify the problem object |
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444 | in this case. |
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445 | |
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446 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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447 | |
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448 | \newpage |
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449 | |
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450 | \section{Basic routines} |
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451 | |
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452 | \subsection{glp\_ios\_reason---determine reason for calling the |
---|
453 | callback routine} |
---|
454 | |
---|
455 | \subsubsection*{Synopsis} |
---|
456 | |
---|
457 | \begin{verbatim} |
---|
458 | int glp_ios_reason(glp_tree *tree); |
---|
459 | \end{verbatim} |
---|
460 | |
---|
461 | \subsubsection*{Returns} |
---|
462 | |
---|
463 | The routine \verb|glp_ios_reason| returns a code, which indicates why |
---|
464 | the user-defined callback routine is being called: |
---|
465 | |
---|
466 | \verb|GLP_ISELECT|---request for subproblem selection; |
---|
467 | |
---|
468 | \verb|GLP_IPREPRO|---request for preprocessing; |
---|
469 | |
---|
470 | \verb|GLP_IROWGEN|---request for row generation; |
---|
471 | |
---|
472 | \verb|GLP_IHEUR |---request for heuristic solution; |
---|
473 | |
---|
474 | \verb|GLP_ICUTGEN|---request for cut generation; |
---|
475 | |
---|
476 | \verb|GLP_IBRANCH|---request for branching; |
---|
477 | |
---|
478 | \verb|GLP_IBINGO |---better integer solution found. |
---|
479 | |
---|
480 | \subsection{glp\_ios\_get\_prob---access the problem object} |
---|
481 | |
---|
482 | \subsubsection*{Synopsis} |
---|
483 | |
---|
484 | \begin{verbatim} |
---|
485 | glp_prob *glp_ios_get_prob(glp_tree *tree); |
---|
486 | \end{verbatim} |
---|
487 | |
---|
488 | \subsubsection*{Description} |
---|
489 | |
---|
490 | The routine \verb|glp_ios_get_prob| can be called from the user-defined |
---|
491 | callback routine to access the problem object, which is used by the MIP |
---|
492 | solver. It is the original problem object passed to the routine |
---|
493 | \verb|glp_intopt| if the MIP presolver is not used; otherwise it is an |
---|
494 | internal problem object built by the presolver. |
---|
495 | |
---|
496 | \subsubsection*{Returns} |
---|
497 | |
---|
498 | The routine \verb|glp_ios_get_prob| returns a pointer to the problem |
---|
499 | object used by the MIP solver. |
---|
500 | |
---|
501 | \subsubsection*{Comments} |
---|
502 | |
---|
503 | To obtain various information about the problem instance the callback |
---|
504 | routine can access the problem object (i.e. the object of type |
---|
505 | \verb|glp_prob|) using the routine \verb|glp_ios_get_prob|. It is the |
---|
506 | original problem object passed to the routine \verb|glp_intopt| if the |
---|
507 | MIP presolver is not used; otherwise it is an internal problem object |
---|
508 | built by the presolver. |
---|
509 | |
---|
510 | \subsection{glp\_ios\_row\_attr---determine additional row attributes} |
---|
511 | |
---|
512 | \subsubsection*{Synopsis} |
---|
513 | |
---|
514 | \begin{verbatim} |
---|
515 | void glp_ios_row_attr(glp_tree *tree, int i, glp_attr *attr); |
---|
516 | \end{verbatim} |
---|
517 | |
---|
518 | \subsubsection*{Description} |
---|
519 | |
---|
520 | The routine \verb|glp_ios_row_attr| retrieves additional attributes of |
---|
521 | $i$-th row of the current subproblem and stores them in the structure |
---|
522 | \verb|glp_attr|, which the parameter \verb|attr| points to. |
---|
523 | |
---|
524 | The structure \verb|glp_attr| has the following fields: |
---|
525 | |
---|
526 | \medskip |
---|
527 | |
---|
528 | \noindent\begin{tabular}{@{}p{17pt}@{}p{120.5mm}@{}} |
---|
529 | \multicolumn{2}{@{}l}{{\tt int level}}\\ |
---|
530 | &Subproblem level at which the row was created. (If \verb|level| = 0, |
---|
531 | the row was added either to the original problem object passed to the |
---|
532 | routine \verb|glp_intopt| or to the root subproblem on generating |
---|
533 | ``lazy'' or/and cutting plane constraints.)\\ |
---|
534 | \end{tabular} |
---|
535 | |
---|
536 | \medskip |
---|
537 | |
---|
538 | \noindent\begin{tabular}{@{}p{17pt}@{}p{120.5mm}@{}} |
---|
539 | \multicolumn{2}{@{}l}{{\tt int origin}}\\ |
---|
540 | &The row origin flag:\\ |
---|
541 | &\verb|GLP_RF_REG |---regular constraint;\\ |
---|
542 | &\verb|GLP_RF_LAZY|---``lazy'' constraint;\\ |
---|
543 | &\verb|GLP_RF_CUT |---cutting plane constraint.\\ |
---|
544 | \end{tabular} |
---|
545 | |
---|
546 | \medskip |
---|
547 | |
---|
548 | \noindent\begin{tabular}{@{}p{17pt}@{}p{120.5mm}@{}} |
---|
549 | \multicolumn{2}{@{}l}{{\tt int klass}}\\ |
---|
550 | &The row class descriptor, which is a number passed to the routine |
---|
551 | \verb|glp_ios_add_row| as its third parameter. If the row is a cutting |
---|
552 | plane constraint generated by the solver, its class may be the |
---|
553 | following:\\ |
---|
554 | &\verb|GLP_RF_GMI |---Gomory's mixed integer cut;\\ |
---|
555 | &\verb|GLP_RF_MIR |---mixed integer rounding cut;\\ |
---|
556 | &\verb|GLP_RF_COV |---mixed cover cut;\\ |
---|
557 | &\verb|GLP_RF_CLQ |---clique cut.\\ |
---|
558 | \end{tabular} |
---|
559 | |
---|
560 | \newpage |
---|
561 | |
---|
562 | \subsection{glp\_ios\_mip\_gap---compute relative MIP gap} |
---|
563 | |
---|
564 | \subsubsection*{Synopsis} |
---|
565 | |
---|
566 | \begin{verbatim} |
---|
567 | double glp_ios_mip_gap(glp_tree *tree); |
---|
568 | \end{verbatim} |
---|
569 | |
---|
570 | \subsubsection*{Description} |
---|
571 | |
---|
572 | The routine \verb|glp_ios_mip_gap| computes the relative MIP gap (also |
---|
573 | called {\it duality gap}) with the following formula: |
---|
574 | $${\tt gap} = \frac{|{\tt best\_mip} - {\tt best\_bnd}|} |
---|
575 | {|{\tt best\_mip}| + {\tt DBL\_EPSILON}}$$ |
---|
576 | where \verb|best_mip| is the best integer feasible solution found so |
---|
577 | far, \verb|best_bnd| is the best (global) bound. If no integer feasible |
---|
578 | solution has been found yet, \verb|gap| is set to \verb|DBL_MAX|. |
---|
579 | |
---|
580 | \subsubsection*{Returns} |
---|
581 | |
---|
582 | The routine \verb|glp_ios_mip_gap| returns the relative MIP gap. |
---|
583 | |
---|
584 | \subsubsection*{Comments} |
---|
585 | |
---|
586 | The relative MIP gap is used to measure the quality of the best integer |
---|
587 | feasible solution found so far, because the optimal solution value |
---|
588 | $z^*$ for the original MIP problem always lies in the range |
---|
589 | $${\tt best\_bnd}\leq z^*\leq{\tt best\_mip}$$ |
---|
590 | in case of minimization, or in the range |
---|
591 | $${\tt best\_mip}\leq z^*\leq{\tt best\_bnd}$$ |
---|
592 | in case of maximization. |
---|
593 | |
---|
594 | To express the relative MIP gap in percents the value returned by the |
---|
595 | routine \verb|glp_ios_mip_gap| should be multiplied by 100\%. |
---|
596 | |
---|
597 | \newpage |
---|
598 | |
---|
599 | \subsection{glp\_ios\_node\_data---access application-specific data} |
---|
600 | |
---|
601 | \subsubsection*{Synopsis} |
---|
602 | |
---|
603 | \begin{verbatim} |
---|
604 | void *glp_ios_node_data(glp_tree *tree, int p); |
---|
605 | \end{verbatim} |
---|
606 | |
---|
607 | \subsubsection*{Description} |
---|
608 | |
---|
609 | The routine \verb|glp_ios_node_data| allows the application accessing a |
---|
610 | memory block allocated for the subproblem (which may be active or |
---|
611 | inactive), whose reference number is $p$. |
---|
612 | |
---|
613 | The size of the block is defined by the control parameter \verb|cb_size| |
---|
614 | passed to the routine \verb|glp_intopt|. The block is initialized by |
---|
615 | binary zeros on creating corresponding subproblem, and its contents is |
---|
616 | kept until the subproblem will be removed from the tree. |
---|
617 | |
---|
618 | The application may use these memory blocks to store specific data for |
---|
619 | each subproblem. |
---|
620 | |
---|
621 | \subsubsection*{Returns} |
---|
622 | |
---|
623 | The routine \verb|glp_ios_node_data| returns a pointer to the memory |
---|
624 | block for the specified subproblem. Note that if \verb|cb_size| = 0, the |
---|
625 | routine returns a null pointer. |
---|
626 | |
---|
627 | \subsection{glp\_ios\_select\_node---select subproblem to continue the |
---|
628 | search} |
---|
629 | |
---|
630 | \subsubsection*{Synopsis} |
---|
631 | |
---|
632 | \begin{verbatim} |
---|
633 | void glp_ios_select_node(glp_tree *tree, int p); |
---|
634 | \end{verbatim} |
---|
635 | |
---|
636 | \subsubsection*{Description} |
---|
637 | |
---|
638 | The routine \verb|glp_ios_select_node| can be called from the |
---|
639 | user-defined callback routine in response to the reason |
---|
640 | \verb|GLP_ISELECT| to select an active subproblem, whose reference |
---|
641 | number is $p$. The search will be continued from the subproblem |
---|
642 | selected. |
---|
643 | |
---|
644 | \newpage |
---|
645 | |
---|
646 | \subsection{glp\_ios\_heur\_sol---provide solution found by heuristic} |
---|
647 | |
---|
648 | \subsubsection*{Synopsis} |
---|
649 | |
---|
650 | \begin{verbatim} |
---|
651 | int glp_ios_heur_sol(glp_tree *tree, const double x[]); |
---|
652 | \end{verbatim} |
---|
653 | |
---|
654 | \subsubsection*{Description} |
---|
655 | |
---|
656 | The routine \verb|glp_ios_heur_sol| can be called from the user-defined |
---|
657 | callback routine in response to the reason \verb|GLP_IHEUR| to provide |
---|
658 | an integer feasible solution found by a primal heuristic. |
---|
659 | |
---|
660 | Primal values of {\it all} variables (columns) found by the heuristic |
---|
661 | should be placed in locations $x[1]$, \dots, $x[n]$, where $n$ is the |
---|
662 | number of columns in the original problem object. Note that the routine |
---|
663 | \verb|glp_ios_heur_sol| does {\it not} check primal feasibility of the |
---|
664 | solution provided. |
---|
665 | |
---|
666 | Using the solution passed in the array $x$ the routine computes value |
---|
667 | of the objective function. If the objective value is better than the |
---|
668 | best known integer feasible solution, the routine computes values of |
---|
669 | auxiliary variables (rows) and stores all solution components in the |
---|
670 | problem object. |
---|
671 | |
---|
672 | \subsubsection*{Returns} |
---|
673 | |
---|
674 | If the provided solution is accepted, the routine |
---|
675 | \verb|glp_ios_heur_sol| returns zero. Otherwise, if the provided |
---|
676 | solution is rejected, the routine returns non-zero. |
---|
677 | |
---|
678 | \subsection{glp\_ios\_can\_branch---check if can branch upon specified |
---|
679 | variable} |
---|
680 | |
---|
681 | \subsubsection*{Synopsis} |
---|
682 | |
---|
683 | \begin{verbatim} |
---|
684 | int glp_ios_can_branch(glp_tree *tree, int j); |
---|
685 | \end{verbatim} |
---|
686 | |
---|
687 | \subsubsection*{Returns} |
---|
688 | |
---|
689 | If $j$-th variable (column) can be used to branch upon, the routine |
---|
690 | returns non-zero, otherwise zero. |
---|
691 | |
---|
692 | \newpage |
---|
693 | |
---|
694 | \subsection{glp\_ios\_branch\_upon---choose variable to branch upon} |
---|
695 | |
---|
696 | \subsubsection*{Synopsis} |
---|
697 | |
---|
698 | \begin{verbatim} |
---|
699 | void glp_ios_branch_upon(glp_tree *tree, int j, int sel); |
---|
700 | \end{verbatim} |
---|
701 | |
---|
702 | \subsubsection*{Description} |
---|
703 | |
---|
704 | The routine \verb|glp_ios_branch_upon| can be called from the |
---|
705 | user-defined callback routine in response to the reason |
---|
706 | \verb|GLP_IBRANCH| to choose a branching variable, whose ordinal number |
---|
707 | is $j$. Should note that only variables, for which the routine |
---|
708 | \verb|glp_ios_can_branch| returns non-zero, can be used to branch upon. |
---|
709 | |
---|
710 | The parameter \verb|sel| is a flag that indicates which branch |
---|
711 | (subproblem) should be selected next to continue the search: |
---|
712 | |
---|
713 | \verb|GLP_DN_BRNCH|---select down-branch; |
---|
714 | |
---|
715 | \verb|GLP_UP_BRNCH|---select up-branch; |
---|
716 | |
---|
717 | \verb|GLP_NO_BRNCH|---use general selection technique. |
---|
718 | |
---|
719 | \subsubsection*{Comments} |
---|
720 | |
---|
721 | On branching the solver removes the current active subproblem from the |
---|
722 | active list and creates two new subproblems ({\it down-} and {\it |
---|
723 | up-branches}), which are added to the end of the active list. Note that |
---|
724 | the down-branch is created before the up-branch, so the last active |
---|
725 | subproblem will be the up-branch. |
---|
726 | |
---|
727 | The down- and up-branches are identical to the current subproblem with |
---|
728 | exception that in the down-branch the upper bound of $x_j$, the variable |
---|
729 | chosen to branch upon, is replaced by $\lfloor x_j^*\rfloor$, while in |
---|
730 | the up-branch the lower bound of $x_j$ is replaced by |
---|
731 | $\lceil x_j^*\rceil$, where $x_j^*$ is the value of $x_j$ in optimal |
---|
732 | solution to LP relaxation of the current subproblem. For example, if |
---|
733 | $x_j^*=3.14$, the new upper bound of $x_j$ in the down-branch is |
---|
734 | $\lfloor 3.14\rfloor=3$, and the new lower bound in the up-branch is |
---|
735 | $\lceil 3.14\rceil=4$.) |
---|
736 | |
---|
737 | Additionally the callback routine may select either down- or up-branch, |
---|
738 | from which the solver will continue the search. If none of the branches |
---|
739 | is selected, a general selection technique will be used. |
---|
740 | |
---|
741 | \newpage |
---|
742 | |
---|
743 | \subsection{glp\_ios\_terminate---terminate the solution process} |
---|
744 | |
---|
745 | \subsubsection*{Synopsis} |
---|
746 | |
---|
747 | \begin{verbatim} |
---|
748 | void glp_ios_terminate(glp_tree *tree); |
---|
749 | \end{verbatim} |
---|
750 | |
---|
751 | \subsubsection*{Description} |
---|
752 | |
---|
753 | The routine \verb|glp_ios_terminate| sets a flag indicating that the |
---|
754 | MIP solver should prematurely terminate the search. |
---|
755 | |
---|
756 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
---|
757 | |
---|
758 | \newpage |
---|
759 | |
---|
760 | \section{The search tree exploring routines} |
---|
761 | |
---|
762 | \subsection{glp\_ios\_tree\_size---determine size of the search tree} |
---|
763 | |
---|
764 | \subsubsection*{Synopsis} |
---|
765 | |
---|
766 | \begin{verbatim} |
---|
767 | void glp_ios_tree_size(glp_tree *tree, int *a_cnt, int *n_cnt, |
---|
768 | int *t_cnt); |
---|
769 | \end{verbatim} |
---|
770 | |
---|
771 | \subsubsection*{Description} |
---|
772 | |
---|
773 | The routine \verb|glp_ios_tree_size| stores the following three counts |
---|
774 | which characterize the current size of the search tree: |
---|
775 | |
---|
776 | \verb|a_cnt| is the current number of active nodes, i.e. the current |
---|
777 | size of the active list; |
---|
778 | |
---|
779 | \verb|n_cnt| is the current number of all (active and inactive) nodes; |
---|
780 | |
---|
781 | \verb|t_cnt| is the total number of nodes including those which have |
---|
782 | been already removed from the tree. This count is increased whenever |
---|
783 | a new node appears in the tree and never decreased. |
---|
784 | |
---|
785 | If some of the parameters \verb|a_cnt|, \verb|n_cnt|, \verb|t_cnt| is |
---|
786 | a null pointer, the corresponding count is not stored. |
---|
787 | |
---|
788 | \subsection{glp\_ios\_curr\_node---determine current active subprob-\\ |
---|
789 | lem} |
---|
790 | |
---|
791 | \subsubsection*{Synopsis} |
---|
792 | |
---|
793 | \begin{verbatim} |
---|
794 | int glp_ios_curr_node(glp_tree *tree); |
---|
795 | \end{verbatim} |
---|
796 | |
---|
797 | \subsubsection*{Returns} |
---|
798 | |
---|
799 | The routine \verb|glp_ios_curr_node| returns the reference number of the |
---|
800 | current active subproblem. However, if the current subproblem does not |
---|
801 | exist, the routine returns zero. |
---|
802 | |
---|
803 | \newpage |
---|
804 | |
---|
805 | \subsection{glp\_ios\_next\_node---determine next active subproblem} |
---|
806 | |
---|
807 | \subsubsection*{Synopsis} |
---|
808 | |
---|
809 | \begin{verbatim} |
---|
810 | int glp_ios_next_node(glp_tree *tree, int p); |
---|
811 | \end{verbatim} |
---|
812 | |
---|
813 | \subsubsection*{Returns} |
---|
814 | |
---|
815 | If the parameter $p$ is zero, the routine \verb|glp_ios_next_node| |
---|
816 | returns the reference number of the first active subproblem. However, |
---|
817 | if the tree is empty, zero is returned. |
---|
818 | |
---|
819 | If the parameter $p$ is not zero, it must specify the reference number |
---|
820 | of some active subproblem, in which case the routine returns the |
---|
821 | reference number of the next active subproblem. However, if there is |
---|
822 | no next active subproblem in the list, zero is returned. |
---|
823 | |
---|
824 | All subproblems in the active list are ordered chronologically, i.e. |
---|
825 | subproblem $A$ precedes subproblem $B$ if $A$ was created before $B$. |
---|
826 | |
---|
827 | \subsection{glp\_ios\_prev\_node---determine previous active subproblem} |
---|
828 | |
---|
829 | \subsubsection*{Synopsis} |
---|
830 | |
---|
831 | \begin{verbatim} |
---|
832 | int glp_ios_prev_node(glp_tree *tree, int p); |
---|
833 | \end{verbatim} |
---|
834 | |
---|
835 | \subsubsection*{Returns} |
---|
836 | |
---|
837 | If the parameter $p$ is zero, the routine \verb|glp_ios_prev_node| |
---|
838 | returns the reference number of the last active subproblem. However, if |
---|
839 | the tree is empty, zero is returned. |
---|
840 | |
---|
841 | If the parameter $p$ is not zero, it must specify the reference number |
---|
842 | of some active subproblem, in which case the routine returns the |
---|
843 | reference number of the previous active subproblem. However, if there |
---|
844 | is no previous active subproblem in the list, zero is returned. |
---|
845 | |
---|
846 | All subproblems in the active list are ordered chronologically, i.e. |
---|
847 | subproblem $A$ precedes subproblem $B$ if $A$ was created before $B$. |
---|
848 | |
---|
849 | \newpage |
---|
850 | |
---|
851 | \subsection{glp\_ios\_up\_node---determine parent subproblem} |
---|
852 | |
---|
853 | \subsubsection*{Synopsis} |
---|
854 | |
---|
855 | \begin{verbatim} |
---|
856 | int glp_ios_up_node(glp_tree *tree, int p); |
---|
857 | \end{verbatim} |
---|
858 | |
---|
859 | \subsubsection*{Returns} |
---|
860 | |
---|
861 | The parameter $p$ must specify the reference number of some (active or |
---|
862 | inactive) subproblem, in which case the routine \verb|iet_get_up_node| |
---|
863 | returns the reference number of its parent subproblem. However, if the |
---|
864 | specified subproblem is the root of the tree and, therefore, has |
---|
865 | no parent, the routine returns zero. |
---|
866 | |
---|
867 | \subsection{glp\_ios\_node\_level---determine subproblem level} |
---|
868 | |
---|
869 | \subsubsection*{Synopsis} |
---|
870 | |
---|
871 | \begin{verbatim} |
---|
872 | int glp_ios_node_level(glp_tree *tree, int p); |
---|
873 | \end{verbatim} |
---|
874 | |
---|
875 | \subsubsection*{Returns} |
---|
876 | |
---|
877 | The routine \verb|glp_ios_node_level| returns the level of the |
---|
878 | subproblem,\linebreak whose reference number is $p$, in the |
---|
879 | branch-and-bound tree. (The root subproblem has level 0, and the level |
---|
880 | of any other subproblem is the level of its parent plus one.) |
---|
881 | |
---|
882 | \subsection{glp\_ios\_node\_bound---determine subproblem local\\bound} |
---|
883 | |
---|
884 | \subsubsection*{Synopsis} |
---|
885 | |
---|
886 | \begin{verbatim} |
---|
887 | double glp_ios_node_bound(glp_tree *tree, int p); |
---|
888 | \end{verbatim} |
---|
889 | |
---|
890 | \subsubsection*{Returns} |
---|
891 | |
---|
892 | The routine \verb|glp_ios_node_bound| returns the local bound for |
---|
893 | (active or inactive) subproblem, whose reference number is $p$. |
---|
894 | |
---|
895 | \subsubsection*{Comments} |
---|
896 | |
---|
897 | The local bound for subproblem $p$ is an lower (minimization) or upper |
---|
898 | (maximization) bound for integer optimal solution to {\it this} |
---|
899 | subproblem (not to the original problem). This bound is local in the |
---|
900 | sense that only subproblems in the subtree rooted at node $p$ cannot |
---|
901 | have better integer feasible solutions. |
---|
902 | |
---|
903 | On creating a subproblem (due to the branching step) its local bound is |
---|
904 | inherited from its parent and then may get only stronger (never weaker). |
---|
905 | For the root subproblem its local bound is initially set to |
---|
906 | \verb|-DBL_MAX| (minimization) or \verb|+DBL_MAX| (maximization) and |
---|
907 | then improved as the root LP relaxation has been solved. |
---|
908 | |
---|
909 | Note that the local bound is not necessarily the optimal objective value |
---|
910 | to corresponding LP relaxation. |
---|
911 | |
---|
912 | \subsection{glp\_ios\_best\_node---find active subproblem with best |
---|
913 | local bound} |
---|
914 | |
---|
915 | \subsubsection*{Synopsis} |
---|
916 | |
---|
917 | \begin{verbatim} |
---|
918 | int glp_ios_best_node(glp_tree *tree); |
---|
919 | \end{verbatim} |
---|
920 | |
---|
921 | \subsubsection*{Returns} |
---|
922 | |
---|
923 | The routine \verb|glp_ios_best_node| returns the reference number of |
---|
924 | the active subproblem, whose local bound is best (i.e. smallest in case |
---|
925 | of minimization or largest in case of maximization). However, if the |
---|
926 | tree is empty, the routine returns zero. |
---|
927 | |
---|
928 | \subsubsection*{Comments} |
---|
929 | |
---|
930 | The best local bound is an lower (minimization) or upper (maximization) |
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931 | bound for integer optimal solution to the original MIP problem. |
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932 | |
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933 | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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934 | |
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935 | \newpage |
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936 | |
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937 | \section{The cut pool routines} |
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938 | |
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939 | \subsection{glp\_ios\_pool\_size---determine current size of the cut\\ |
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940 | pool} |
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941 | |
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942 | \subsubsection*{Synopsis} |
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943 | |
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944 | \begin{verbatim} |
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945 | int glp_ios_pool_size(glp_tree *tree); |
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946 | \end{verbatim} |
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947 | |
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948 | \subsubsection*{Returns} |
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949 | |
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950 | The routine \verb|glp_ios_pool_size| returns the current size of the |
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951 | cut pool, that is, the number of cutting plane constraints currently |
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952 | added to it. |
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953 | |
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954 | \subsection{glp\_ios\_add\_row---add constraint to the cut pool} |
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955 | |
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956 | \subsubsection*{Synopsis} |
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957 | |
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958 | \begin{verbatim} |
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959 | int glp_ios_add_row(glp_tree *tree, const char *name, |
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960 | int klass, int flags, int len, const int ind[], |
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961 | const double val[], int type, double rhs); |
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962 | \end{verbatim} |
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963 | |
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964 | \subsubsection*{Description} |
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965 | |
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966 | The routine \verb|glp_ios_add_row| adds specified row (cutting plane |
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967 | constraint) to the cut pool. |
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968 | |
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969 | The cutting plane constraint should have the following format: |
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970 | $$\sum_{j\in J}a_jx_j\left\{\begin{array}{@{}c@{}}\geq\\\leq\\ |
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971 | \end{array}\right\}b,$$ |
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972 | where $J$ is a set of indices (ordinal numbers) of structural variables, |
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973 | $a_j$ are constraint coefficients, $x_j$ are structural variables, $b$ |
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974 | is the right-hand side. |
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975 | |
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976 | The parameter \verb|name| specifies a symbolic name assigned to the |
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977 | constraint (1 up to 255 characters). If it is \verb|NULL| or an empty |
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978 | string, no name is assigned. |
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979 | |
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980 | The parameter \verb|klass| specifies the constraint class, which must |
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981 | be either zero or a number in the range from 101 to 200. The application |
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982 | may use this attribute to distinguish between cutting plane constraints |
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983 | of different classes.\footnote{Constraint classes numbered from 1 to 100 |
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984 | are reserved for GLPK cutting plane generators.} |
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985 | |
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986 | The parameter \verb|flags| currently is not used and must be zero. |
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987 | |
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988 | Ordinal numbers of structural variables (i.e. column indices) $j\in J$ |
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989 | and numerical values of corresponding constraint coefficients $a_j$ must |
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990 | be placed in locations \verb|ind[1]|, \dots, \verb|ind[len]| and |
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991 | \verb|val[1]|, \dots, \verb|val[len]|, respectively, where |
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992 | ${\tt len}=|J|$ is the number of constraint coefficients, |
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993 | $0\leq{\tt len}\leq n$, and $n$ is the number of columns in the problem |
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994 | object. Coefficients with identical column indices are not allowed. |
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995 | Zero coefficients are allowed, however, they are ignored. |
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996 | |
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997 | The parameter \verb|type| specifies the constraint type as follows: |
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998 | |
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999 | \verb|GLP_LO| means inequality constraint $\Sigma a_jx_j\geq b$; |
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1000 | |
---|
1001 | \verb|GLP_UP| means inequality constraint $\Sigma a_jx_j\leq b$; |
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1002 | |
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1003 | The parameter \verb|rhs| specifies the right-hand side $b$. |
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1004 | |
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1005 | All cutting plane constraints in the cut pool are identified by their |
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1006 | ordinal numbers 1, 2, \dots, $size$, where $size$ is the current size |
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1007 | of the cut pool. New constraints are always added to the end of the cut |
---|
1008 | pool, thus, ordinal numbers of previously added constraints are not |
---|
1009 | changed. |
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1010 | |
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1011 | \subsubsection*{Returns} |
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1012 | |
---|
1013 | The routine \verb|glp_ios_add_row| returns the ordinal number of the |
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1014 | cutting plane constraint added, which is the new size of the cut pool. |
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1015 | |
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1016 | \subsubsection*{Example} |
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1017 | |
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1018 | \begin{verbatim} |
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1019 | /* generate triangle cutting plane: |
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1020 | x[i] + x[j] + x[k] <= 1 */ |
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1021 | . . . |
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1022 | /* add the constraint to the cut pool */ |
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1023 | ind[1] = i, val[1] = 1.0; |
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1024 | ind[2] = j, val[2] = 1.0; |
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1025 | ind[3] = k, val[3] = 1.0; |
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1026 | glp_ios_add_row(tree, NULL, TRIANGLE_CUT, 0, 3, ind, val, |
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1027 | GLP_UP, 1.0); |
---|
1028 | \end{verbatim} |
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1029 | |
---|
1030 | \subsubsection*{Comments} |
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1031 | |
---|
1032 | Cutting plane constraints added to the cut pool are intended to be then |
---|
1033 | added only to the {\it current} subproblem, so these constraints can be |
---|
1034 | globally as well as locally valid. However, adding a constraint to the |
---|
1035 | cut pool does not mean that it will be added to the current |
---|
1036 | subproblem---it depends on the solver's decision: if the constraint |
---|
1037 | seems to be efficient, it is moved from the pool to the current |
---|
1038 | subproblem, otherwise it is simply dropped.\footnote{Globally valid |
---|
1039 | constraints could be saved and then re-used for other subproblems, but |
---|
1040 | currently such feature is not implemented.} |
---|
1041 | |
---|
1042 | Normally, every time the callback routine is called for cut generation, |
---|
1043 | the cut pool is empty. On the other hand, the solver itself can generate |
---|
1044 | cutting plane constraints (like Gomory's or mixed integer rounding |
---|
1045 | cuts), in which case the cut pool may be non-empty. |
---|
1046 | |
---|
1047 | \subsection{glp\_ios\_del\_row---remove constraint from the cut pool} |
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1048 | |
---|
1049 | \subsubsection*{Synopsis} |
---|
1050 | |
---|
1051 | \begin{verbatim} |
---|
1052 | void glp_ios_del_row(glp_tree *tree, int i); |
---|
1053 | \end{verbatim} |
---|
1054 | |
---|
1055 | \subsubsection*{Description} |
---|
1056 | |
---|
1057 | The routine \verb|glp_ios_del_row| deletes $i$-th row (cutting plane |
---|
1058 | constraint) from the cut pool, where $1\leq i\leq size$ is the ordinal |
---|
1059 | number of the constraint in the pool, $size$ is the current size of the |
---|
1060 | cut pool. |
---|
1061 | |
---|
1062 | Note that deleting a constraint from the cut pool leads to changing |
---|
1063 | ordinal numbers of other constraints remaining in the pool. New ordinal |
---|
1064 | numbers of the remaining constraints are assigned under assumption that |
---|
1065 | the original order of constraints is not changed. Let, for example, |
---|
1066 | there be four constraints $a$, $b$, $c$ and $d$ in the cut pool, which |
---|
1067 | have ordinal numbers 1, 2, 3 and 4, respectively, and let constraint |
---|
1068 | $b$ have been deleted. Then after deletion the remaining constraint $a$, |
---|
1069 | $c$ and $d$ are assigned new ordinal numbers 1, 2 and 3, respectively. |
---|
1070 | |
---|
1071 | To find the constraint to be deleted the routine \verb|glp_ios_del_row| |
---|
1072 | uses ``smart'' linear search, so it is recommended to remove constraints |
---|
1073 | in a natural or reverse order and avoid removing them in a random order. |
---|
1074 | |
---|
1075 | \subsubsection*{Example} |
---|
1076 | |
---|
1077 | \begin{verbatim} |
---|
1078 | /* keep first 10 constraints in the cut pool and remove other |
---|
1079 | constraints */ |
---|
1080 | while (glp_ios_pool_size(tree) > 10) |
---|
1081 | glp_ios_del_row(tree, glp_ios_pool_size(tree)); |
---|
1082 | \end{verbatim} |
---|
1083 | |
---|
1084 | \newpage |
---|
1085 | |
---|
1086 | \subsection{glp\_ios\_clear\_pool---remove all constraints from the cut |
---|
1087 | pool} |
---|
1088 | |
---|
1089 | \subsubsection*{Synopsis} |
---|
1090 | |
---|
1091 | \begin{verbatim} |
---|
1092 | void glp_ios_clear_pool(glp_tree *tree); |
---|
1093 | \end{verbatim} |
---|
1094 | |
---|
1095 | \subsubsection*{Description} |
---|
1096 | |
---|
1097 | The routine \verb|glp_ios_clear_pool| makes the cut pool empty deleting |
---|
1098 | all existing rows (cutting plane constraints) from it. |
---|
1099 | |
---|
1100 | %* eof *% |
---|