1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/doc/glpk05.tex Mon Dec 06 13:09:21 2010 +0100
1.3 @@ -0,0 +1,1100 @@
1.4 +%* glpk05.tex *%
1.5 +
1.6 +\chapter{Branch-and-Cut API Routines}
1.7 +
1.8 +\section{Introduction}
1.9 +
1.10 +\subsection{Using the callback routine}
1.11 +
1.12 +The GLPK MIP solver based on the branch-and-cut method allows the
1.13 +application program to control the solution process. This is attained
1.14 +by means of the user-defined callback routine, which is called by the
1.15 +solver at various points of the branch-and-cut algorithm.
1.16 +
1.17 +The callback routine passed to the MIP solver should be written by the
1.18 +user and has the following specification:\footnote{The name
1.19 +{\tt foo\_bar} used here is a placeholder for the callback routine
1.20 +name.}
1.21 +
1.22 +\begin{verbatim}
1.23 + void foo_bar(glp_tree *tree, void *info);
1.24 +\end{verbatim}
1.25 +
1.26 +\noindent
1.27 +where \verb|tree| is a pointer to the data structure \verb|glp_tree|,
1.28 +which should be used on subsequent calls to branch-and-cut interface
1.29 +routines, and \verb|info| is a transit pointer passed to the routine
1.30 +\verb|glp_intopt|, which may be used by the application program to pass
1.31 +some external data to the callback routine.
1.32 +
1.33 +The callback routine is passed to the MIP solver through the control
1.34 +parameter structure \verb|glp_iocp| (see Chapter ``Basic API Routines'',
1.35 +Section ``Mixed integer programming routines'', Subsection ``Solve MIP
1.36 +problem with the branch-and-cut method'') as follows:
1.37 +
1.38 +\newpage
1.39 +
1.40 +\begin{verbatim}
1.41 + glp_prob *mip;
1.42 + glp_iocp parm;
1.43 + . . .
1.44 + glp_init_iocp(&parm);
1.45 + . . .
1.46 + parm.cb_func = foo_bar;
1.47 + parm.cb_info = ... ;
1.48 + ret = glp_intopt(mip, &parm);
1.49 + . . .
1.50 +\end{verbatim}
1.51 +
1.52 +To determine why it is being called by the MIP solver the callback
1.53 +routine should use the routine \verb|glp_ios_reason| (described in this
1.54 +section below), which returns a code indicating the reason for calling.
1.55 +Depending on the reason the callback routine may perform necessary
1.56 +actions to control the solution process.
1.57 +
1.58 +The reason codes, which correspond to various point of the
1.59 +branch-and-cut algorithm implemented in the MIP solver, are described
1.60 +in Subsection ``Reasons for calling the callback routine'' below.
1.61 +
1.62 +To ignore calls for reasons, which are not processed by the callback
1.63 +routine, it should just return to the MIP solver doing nothing. For
1.64 +example:
1.65 +
1.66 +\begin{verbatim}
1.67 +void foo_bar(glp_tree *tree, void *info)
1.68 +{ . . .
1.69 + switch (glp_ios_reason(tree))
1.70 + { case GLP_IBRANCH:
1.71 + . . .
1.72 + break;
1.73 + case GLP_ISELECT:
1.74 + . . .
1.75 + break;
1.76 + default:
1.77 + /* ignore call for other reasons */
1.78 + break;
1.79 + }
1.80 + return;
1.81 +}
1.82 +\end{verbatim}
1.83 +
1.84 +To control the solution process as well as to obtain necessary
1.85 +information the callback routine may use the branch-and-cut API
1.86 +routines described in this chapter. Names of all these routines begin
1.87 +with `\verb|glp_ios_|'.
1.88 +
1.89 +\subsection{Branch-and-cut algorithm}
1.90 +
1.91 +This section gives a schematic description of the branch-and-cut
1.92 +algorithm as it is implemented in the GLPK MIP solver.
1.93 +
1.94 +\medskip
1.95 +
1.96 +{\it 1. Initialization}
1.97 +
1.98 +Set $L:=\{P_0\}$, where $L$ is the {\it active list} (i.e. the list of
1.99 +active subproblems), $P_0$ is the original MIP problem to be solved.
1.100 +
1.101 +Set $z^{\it best}:=+\infty$ (in case of minimization) or
1.102 +$z^{\it best}:=-\infty$ (in case of maximization), where $z^{\it best}$
1.103 +is {\it incumbent value}, i.e. an upper (minimization) or lower
1.104 +(maximization) global bound for $z^{\it opt}$, the optimal objective
1.105 +value for $P^0$.
1.106 +
1.107 +\medskip
1.108 +
1.109 +{\it 2. Subproblem selection}
1.110 +
1.111 +If $L=\varnothing$ then GO TO 9.
1.112 +
1.113 +Select $P\in L$, i.e. make active subproblem $P$ current.
1.114 +
1.115 +\medskip
1.116 +
1.117 +{\it 3. Solving LP relaxation}
1.118 +
1.119 +Solve $P^{\it LP}$, which is LP relaxation of $P$.
1.120 +
1.121 +If $P^{\it LP}$ has no primal feasible solution then GO TO 8.
1.122 +
1.123 +Let $z^{\it LP}$ be the optimal objective value for $P^{\it LP}$.
1.124 +
1.125 +If $z^{\it LP}\geq z^{\it best}$ (in case of minimization) or
1.126 +$z^{\it LP}\leq z^{\rm best}$ (in case of maximization) then GO TO 8.
1.127 +
1.128 +\medskip
1.129 +
1.130 +{\it 4. Adding ``lazy'' constraints}
1.131 +
1.132 +Let $x^{\it LP}$ be the optimal solution to $P^{\it LP}$.
1.133 +
1.134 +If there are ``lazy'' constraints (i.e. essential constraints not
1.135 +included in the original MIP problem $P_0$), which are violated at the
1.136 +optimal point $x^{\it LP}$, add them to $P$, and GO TO 3.
1.137 +
1.138 +\medskip
1.139 +
1.140 +{\it 5. Check for integrality}
1.141 +
1.142 +Let $x_j$ be a variable, which is required to be integer, and let
1.143 +$x^{\it LP}_j\in x^{\it LP}$ be its value in the optimal solution to
1.144 +$P^{\it LP}$.
1.145 +
1.146 +If $x^{\it LP}_j$ are integral for all integer variables, then a better
1.147 +integer feasible solution is found. Store its components, set
1.148 +$z^{\it best}:=z^{\it LP}$, and GO TO 8.
1.149 +
1.150 +\medskip
1.151 +
1.152 +{\it 6. Adding cutting planes}
1.153 +
1.154 +If there are cutting planes (i.e. valid constraints for $P$),
1.155 +which are violated at the optimal point $x^{\it LP}$, add them to $P$,
1.156 +and GO TO 3.
1.157 +
1.158 +\medskip
1.159 +
1.160 +{\it 7. Branching}
1.161 +
1.162 +Select {\it branching variable} $x_j$, i.e. a variable, which is
1.163 +required to be integer, and whose value $x^{\it LP}_j\in x^{\it LP}$ is
1.164 +fractional in the optimal solution to $P^{\it LP}$.
1.165 +
1.166 +Create new subproblem $P^D$ (so called {\it down branch}), which is
1.167 +identical to the current subproblem $P$ with exception that the upper
1.168 +bound of $x_j$ is replaced by $\lfloor x^{\it LP}_j\rfloor$. (For
1.169 +example, if $x^{\it LP}_j=3.14$, the new upper bound of $x_j$ in the
1.170 +down branch will be $\lfloor 3.14\rfloor=3$.)
1.171 +
1.172 +Create new subproblem $P^U$ (so called {\it up branch}), which is
1.173 +identical to the current subproblem $P$ with exception that the lower
1.174 +bound of $x_j$ is replaced by $\lceil x^{\it LP}_j\rceil$. (For example,
1.175 +if $x^{\it LP}_j=3.14$, the new lower bound of $x_j$ in the up branch
1.176 +will be $\lceil 3.14\rceil=4$.)
1.177 +
1.178 +Set $L:=(L\backslash\{P\})\cup\{P^D,P^U\}$, i.e. remove the current
1.179 +subproblem $P$ from the active list $L$ and add two new subproblems
1.180 +$P^D$ and $P^U$ to it. Then GO TO 2.
1.181 +
1.182 +\medskip
1.183 +
1.184 +{\it 8. Pruning}
1.185 +
1.186 +Remove from the active list $L$ all subproblems (including the current
1.187 +one), whose local bound $\widetilde{z}$ is not better than the global
1.188 +bound $z^{\it best}$, i.e. set $L:=L\backslash\{P\}$ for all $P$, where
1.189 +$\widetilde{z}\geq z^{\it best}$ (in case of minimization) or
1.190 +$\widetilde{z}\leq z^{\it best}$ (in case of maximization), and then
1.191 +GO TO 2.
1.192 +
1.193 +The local bound $\widetilde{z}$ for subproblem $P$ is an lower
1.194 +(minimization) or upper (maximization) bound for integer optimal
1.195 +solution to {\it this} subproblem (not to the original problem). This
1.196 +bound is local in the sense that only subproblems in the subtree rooted
1.197 +at node $P$ cannot have better integer feasible solutions. Note that
1.198 +the local bound is not necessarily the optimal objective value to LP
1.199 +relaxation $P^{\it LP}$.
1.200 +
1.201 +\medskip
1.202 +
1.203 +{\it 9. Termination}
1.204 +
1.205 +If $z^{\it best}=+\infty$ (in case of minimization) or
1.206 +$z^{\it best}=-\infty$ (in case of maximization), the original problem
1.207 +$P_0$ has no integer feasible solution. Otherwise, the last integer
1.208 +feasible solution stored on step 5 is the integer optimal solution to
1.209 +the original problem $P_0$ with $z^{\it opt}=z^{\it best}$. STOP.
1.210 +
1.211 +\subsection{The search tree}
1.212 +
1.213 +On the branching step of the branch-and-cut algorithm the current
1.214 +subproblem is divided into two\footnote{In more general cases the
1.215 +current subproblem may be divided into more than two subproblems.
1.216 +However, currently such feature is not used in GLPK.} new subproblems,
1.217 +so the set of all subproblems can be represented in the form of a rooted
1.218 +tree, which is called the {\it search} or {\it branch-and-bound} tree.
1.219 +An example of the search tree is shown on Fig.~1. Each node of the
1.220 +search tree corresponds to a subproblem, so the terms `node' and
1.221 +`subproblem' may be used synonymously.
1.222 +
1.223 +\newpage
1.224 +
1.225 +\begin{figure}[t]
1.226 +\noindent\hfil
1.227 +\xymatrix @R=20pt @C=10pt
1.228 +{&&&&&&*+<14pt>[o][F=]{A}\ar@{-}[dllll]\ar@{-}[dr]\ar@{-}[drrrr]&&&&\\
1.229 +&&*+<14pt>[o][F=]{B}\ar@{-}[dl]\ar@{-}[dr]&&&&&*+<14pt>[o][F=]{C}
1.230 +\ar@{-}[dll]\ar@{-}[dr]\ar@{-}[drrr]&&&*+<14pt>[o][F-]{\times}\\
1.231 +&*+<14pt>[o][F-]{\times}\ar@{-}[dl]\ar@{-}[d]\ar@{-}[dr]&&
1.232 +*+<14pt>[o][F-]{D}&&*+<14pt>[o][F=]{E}\ar@{-}[dl]\ar@{-}[dr]&&&
1.233 +*+<14pt>[o][F=]{F}\ar@{-}[dl]\ar@{-}[dr]&&*+<14pt>[o][F-]{G}\\
1.234 +*+<14pt>[o][F-]{\times}&*+<14pt>[o][F-]{\times}&*+<14pt>[o][F-]{\times}
1.235 +&&*+<14pt>[][F-]{H}&&*+<14pt>[o][F-]{I}&*+<14pt>[o][F-]{\times}&&
1.236 +*+<14pt>[o][F-]{J}&\\}
1.237 +
1.238 +\bigskip
1.239 +
1.240 +\noindent\hspace{.8in}
1.241 +\xymatrix @R=11pt
1.242 +{*+<20pt>[][F-]{}&*\txt{\makebox[1in][l]{Current}}&&
1.243 +*+<20pt>[o][F-]{}&*\txt{\makebox[1in][l]{Active}}\\
1.244 +*+<20pt>[o][F=]{}&*\txt{\makebox[1in][l]{Non-active}}&&
1.245 +*+<14pt>[o][F-]{\times}&*\txt{\makebox[1in][l]{Fathomed}}\\
1.246 +}
1.247 +
1.248 +\begin{center}
1.249 +Fig. 1. An example of the search tree.
1.250 +\end{center}
1.251 +\end{figure}
1.252 +
1.253 +In GLPK each node may have one of the following four statuses:
1.254 +
1.255 +$\bullet$ {\it current node} is the active node currently being
1.256 +processed;
1.257 +
1.258 +$\bullet$ {\it active node} is a leaf node, which still has to be
1.259 +processed;
1.260 +
1.261 +$\bullet$ {\it non-active node} is a node, which has been processed,
1.262 +but not fathomed;
1.263 +
1.264 +$\bullet$ {\it fathomed node} is a node, which has been processed and
1.265 +fathomed.
1.266 +
1.267 +In the data structure representing the search tree GLPK keeps only
1.268 +current, active, and non-active nodes. Once a node has been fathomed,
1.269 +it is removed from the tree data structure.
1.270 +
1.271 +Being created each node of the search tree is assigned a distinct
1.272 +positive integer called the {\it subproblem reference number}, which
1.273 +may be used by the application program to specify a particular node of
1.274 +the tree. The root node corresponding to the original problem to be
1.275 +solved is always assigned the reference number 1.
1.276 +
1.277 +\subsection{Current subproblem}
1.278 +
1.279 +The current subproblem is a MIP problem corresponding to the current
1.280 +node of the search tree. It is represented as the GLPK problem object
1.281 +(\verb|glp_prob|) that allows the application program using API routines
1.282 +to access its content in the standard way. If the MIP presolver is not
1.283 +used, it is the original problem object passed to the routine
1.284 +\verb|glp_intopt|; otherwise, it is an internal problem object built by
1.285 +the MIP presolver.
1.286 +
1.287 +Note that the problem object is used by the MIP solver itself during
1.288 +the solution process for various purposes (to solve LP relaxations, to
1.289 +perfom branching, etc.), and even if the MIP presolver is not used, the
1.290 +current content of the problem object may differ from its original
1.291 +content. For example, it may have additional rows, bounds of some rows
1.292 +and columns may be changed, etc. In particular, LP segment of the
1.293 +problem object corresponds to LP relaxation of the current subproblem.
1.294 +However, on exit from the MIP solver the content of the problem object
1.295 +is restored to its original state.
1.296 +
1.297 +To obtain information from the problem object the application program
1.298 +may use any API routines, which do not change the object. Using API
1.299 +routines, which change the problem object, is restricted to stipulated
1.300 +cases.
1.301 +
1.302 +\subsection{The cut pool}
1.303 +
1.304 +The {\it cut pool} is a set of cutting plane constraints maintained by
1.305 +the MIP solver. It is used by the GLPK cut generation routines and may
1.306 +be used by the application program in the same way, i.e. rather than
1.307 +to add cutting plane constraints directly to the problem object the
1.308 +application program may store them to the cut pool. In the latter case
1.309 +the solver looks through the cut pool, selects efficient constraints,
1.310 +and adds them to the problem object.
1.311 +
1.312 +\subsection{Reasons for calling the callback routine}
1.313 +
1.314 +The callback routine may be called by the MIP solver for the following
1.315 +reasons.
1.316 +
1.317 +\subsubsection*{Request for subproblem selection}
1.318 +
1.319 +The callback routine is called with the reason code \verb|GLP_ISELECT|
1.320 +if the current subproblem has been fathomed and therefore there is no
1.321 +current subproblem.
1.322 +
1.323 +In response the callback routine may select some subproblem from the
1.324 +active list and pass its reference number to the solver using the
1.325 +routine \verb|glp_ios_select_node|, in which case the solver continues
1.326 +the search from the specified active subproblem. If no selection is made
1.327 +by the callback routine, the solver uses a backtracking technique
1.328 +specified by the control parameter \verb|bt_tech|.
1.329 +
1.330 +To explore the active list (i.e. active nodes of the branch-and-bound
1.331 +tree) the callback routine may use the routines \verb|glp_ios_next_node|
1.332 +and \verb|glp_ios_prev_node|.
1.333 +
1.334 +\subsubsection*{Request for preprocessing}
1.335 +
1.336 +The callback routine is called with the reason code \verb|GLP_IPREPRO|
1.337 +if the current subproblem has just been selected from the active list
1.338 +and its LP relaxation is not solved yet.
1.339 +
1.340 +In response the callback routine may perform some preprocessing of the
1.341 +current subproblem like tightening bounds of some variables or removing
1.342 +bounds of some redundant constraints.
1.343 +
1.344 +\subsubsection*{Request for row generation}
1.345 +
1.346 +The callback routine is called with the reason code \verb|GLP_IROWGEN|
1.347 +if LP relaxation of the current subproblem has just been solved to
1.348 +optimality and its objective value is better than the best known integer
1.349 +feasible solution.
1.350 +
1.351 +In response the callback routine may add one or more ``lazy''
1.352 +constraints (rows), which are violated by the current optimal solution
1.353 +of LP relaxation, using API routines \verb|glp_add_rows|,
1.354 +\verb|glp_set_row_name|, \verb|glp_set_row_bnds|, and
1.355 +\verb|glp_set_mat_row|, in which case the solver will perform
1.356 +re-optimization of LP relaxation. If there are no violated constraints,
1.357 +the callback routine should just return.
1.358 +
1.359 +Optimal solution components for LP relaxation can be obtained with API
1.360 +routines \verb|glp_get_obj_val|, \verb|glp_get_row_prim|,
1.361 +\verb|glp_get_row_dual|, \verb|glp_get_col_prim|, and
1.362 +\verb|glp_get_col_dual|.
1.363 +
1.364 +\subsubsection*{Request for heuristic solution}
1.365 +
1.366 +The callback routine is called with the reason code \verb|GLP_IHEUR|
1.367 +if LP relaxation of the current subproblem being solved to optimality
1.368 +is integer infeasible (i.e. values of some structural variables of
1.369 +integer kind are fractional), though its objective value is better than
1.370 +the best known integer feasible solution.
1.371 +
1.372 +In response the callback routine may try applying a primal heuristic
1.373 +to find an integer feasible solution,\footnote{Integer feasible to the
1.374 +original MIP problem, not to the current subproblem.} which is better
1.375 +than the best known one. In case of success the callback routine may
1.376 +store such better solution in the problem object using the routine
1.377 +\verb|glp_ios_heur_sol|.
1.378 +
1.379 +\subsubsection*{Request for cut generation}
1.380 +
1.381 +The callback routine is called with the reason code \verb|GLP_ICUTGEN|
1.382 +if LP relaxation of the current subproblem being solved to optimality
1.383 +is integer infeasible (i.e. values of some structural variables of
1.384 +integer kind are fractional), though its objective value is better than
1.385 +the best known integer feasible solution.
1.386 +
1.387 +In response the callback routine may reformulate the {\it current}
1.388 +subproblem (before it will be splitted up due to branching) by adding to
1.389 +the problem object one or more {\it cutting plane constraints}, which
1.390 +cut off the fractional optimal point from the MIP
1.391 +polytope.\footnote{Since these constraints are added to the current
1.392 +subproblem, they may be globally as well as locally valid.}
1.393 +
1.394 +Adding cutting plane constraints may be performed in two ways.
1.395 +One way is the same as for the reason code \verb|GLP_IROWGEN| (see
1.396 +above), in which case the callback routine adds new rows corresponding
1.397 +to cutting plane constraints directly to the current subproblem.
1.398 +
1.399 +The other way is to add cutting plane constraints to the {\it cut pool},
1.400 +a set of cutting plane constraints maintained by the solver, rather than
1.401 +directly to the current subproblem. In this case after return from the
1.402 +callback routine the solver looks through the cut pool, selects
1.403 +efficient cutting plane constraints, adds them to the current
1.404 +subproblem, drops other constraints, and then performs re-optimization.
1.405 +
1.406 +\subsubsection*{Request for branching}
1.407 +
1.408 +The callback routine is called with the reason code \verb|GLP_IBRANCH|
1.409 +if LP relaxation of the current subproblem being solved to optimality
1.410 +is integer infeasible (i.e. values of some structural variables of
1.411 +integer kind are fractional), though its objective value is better than
1.412 +the best known integer feasible solution.
1.413 +
1.414 +In response the callback routine may choose some variable suitable for
1.415 +branching (i.e. integer variable, whose value in optimal solution to
1.416 +LP relaxation of the current subproblem is fractional) and pass its
1.417 +ordinal number to the solver using the routine
1.418 +\verb|glp_ios_branch_upon|, in which case the solver splits the current
1.419 +subproblem in two new subproblems and continues the search. If no choice
1.420 +is made by the callback routine, the solver uses a branching technique
1.421 +specified by the control parameter \verb|br_tech|.
1.422 +
1.423 +\subsubsection*{Better integer solution found}
1.424 +
1.425 +The callback routine is called with the reason code \verb|GLP_IBINGO|
1.426 +if LP relaxation of the current subproblem being solved to optimality
1.427 +is integer feasible (i.e. values of all structural variables of integer
1.428 +kind are integral within the working precision) and its objective value
1.429 +is better than the best known integer feasible solution.
1.430 +
1.431 +Optimal solution components for LP relaxation can be obtained in the
1.432 +same way as for the reason code \verb|GLP_IROWGEN| (see above).
1.433 +
1.434 +Components of the new MIP solution can be obtained with API routines
1.435 +\verb|glp_mip_obj_val|, \verb|glp_mip_row_val|, and
1.436 +\verb|glp_mip_col_val|. Note, however, that due to row/cut generation
1.437 +there may be additional rows in the problem object.
1.438 +
1.439 +The difference between optimal solution to LP relaxation and
1.440 +corresponding MIP solution is that in the former case some structural
1.441 +variables of integer kind (namely, basic variables) may have values,
1.442 +which are close to nearest integers within the working precision, while
1.443 +in the latter case all such variables have exact integral values.
1.444 +
1.445 +The reason \verb|GLP_IBINGO| is intended only for informational
1.446 +purposes, so the callback routine should not modify the problem object
1.447 +in this case.
1.448 +
1.449 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1.450 +
1.451 +\newpage
1.452 +
1.453 +\section{Basic routines}
1.454 +
1.455 +\subsection{glp\_ios\_reason---determine reason for calling the
1.456 +callback routine}
1.457 +
1.458 +\subsubsection*{Synopsis}
1.459 +
1.460 +\begin{verbatim}
1.461 +int glp_ios_reason(glp_tree *tree);
1.462 +\end{verbatim}
1.463 +
1.464 +\subsubsection*{Returns}
1.465 +
1.466 +The routine \verb|glp_ios_reason| returns a code, which indicates why
1.467 +the user-defined callback routine is being called:
1.468 +
1.469 +\verb|GLP_ISELECT|---request for subproblem selection;
1.470 +
1.471 +\verb|GLP_IPREPRO|---request for preprocessing;
1.472 +
1.473 +\verb|GLP_IROWGEN|---request for row generation;
1.474 +
1.475 +\verb|GLP_IHEUR |---request for heuristic solution;
1.476 +
1.477 +\verb|GLP_ICUTGEN|---request for cut generation;
1.478 +
1.479 +\verb|GLP_IBRANCH|---request for branching;
1.480 +
1.481 +\verb|GLP_IBINGO |---better integer solution found.
1.482 +
1.483 +\subsection{glp\_ios\_get\_prob---access the problem object}
1.484 +
1.485 +\subsubsection*{Synopsis}
1.486 +
1.487 +\begin{verbatim}
1.488 +glp_prob *glp_ios_get_prob(glp_tree *tree);
1.489 +\end{verbatim}
1.490 +
1.491 +\subsubsection*{Description}
1.492 +
1.493 +The routine \verb|glp_ios_get_prob| can be called from the user-defined
1.494 +callback routine to access the problem object, which is used by the MIP
1.495 +solver. It is the original problem object passed to the routine
1.496 +\verb|glp_intopt| if the MIP presolver is not used; otherwise it is an
1.497 +internal problem object built by the presolver.
1.498 +
1.499 +\subsubsection*{Returns}
1.500 +
1.501 +The routine \verb|glp_ios_get_prob| returns a pointer to the problem
1.502 +object used by the MIP solver.
1.503 +
1.504 +\subsubsection*{Comments}
1.505 +
1.506 +To obtain various information about the problem instance the callback
1.507 +routine can access the problem object (i.e. the object of type
1.508 +\verb|glp_prob|) using the routine \verb|glp_ios_get_prob|. It is the
1.509 +original problem object passed to the routine \verb|glp_intopt| if the
1.510 +MIP presolver is not used; otherwise it is an internal problem object
1.511 +built by the presolver.
1.512 +
1.513 +\subsection{glp\_ios\_row\_attr---determine additional row attributes}
1.514 +
1.515 +\subsubsection*{Synopsis}
1.516 +
1.517 +\begin{verbatim}
1.518 +void glp_ios_row_attr(glp_tree *tree, int i, glp_attr *attr);
1.519 +\end{verbatim}
1.520 +
1.521 +\subsubsection*{Description}
1.522 +
1.523 +The routine \verb|glp_ios_row_attr| retrieves additional attributes of
1.524 +$i$-th row of the current subproblem and stores them in the structure
1.525 +\verb|glp_attr|, which the parameter \verb|attr| points to.
1.526 +
1.527 +The structure \verb|glp_attr| has the following fields:
1.528 +
1.529 +\medskip
1.530 +
1.531 +\noindent\begin{tabular}{@{}p{17pt}@{}p{120.5mm}@{}}
1.532 +\multicolumn{2}{@{}l}{{\tt int level}}\\
1.533 +&Subproblem level at which the row was created. (If \verb|level| = 0,
1.534 +the row was added either to the original problem object passed to the
1.535 +routine \verb|glp_intopt| or to the root subproblem on generating
1.536 +``lazy'' or/and cutting plane constraints.)\\
1.537 +\end{tabular}
1.538 +
1.539 +\medskip
1.540 +
1.541 +\noindent\begin{tabular}{@{}p{17pt}@{}p{120.5mm}@{}}
1.542 +\multicolumn{2}{@{}l}{{\tt int origin}}\\
1.543 +&The row origin flag:\\
1.544 +&\verb|GLP_RF_REG |---regular constraint;\\
1.545 +&\verb|GLP_RF_LAZY|---``lazy'' constraint;\\
1.546 +&\verb|GLP_RF_CUT |---cutting plane constraint.\\
1.547 +\end{tabular}
1.548 +
1.549 +\medskip
1.550 +
1.551 +\noindent\begin{tabular}{@{}p{17pt}@{}p{120.5mm}@{}}
1.552 +\multicolumn{2}{@{}l}{{\tt int klass}}\\
1.553 +&The row class descriptor, which is a number passed to the routine
1.554 +\verb|glp_ios_add_row| as its third parameter. If the row is a cutting
1.555 +plane constraint generated by the solver, its class may be the
1.556 +following:\\
1.557 +&\verb|GLP_RF_GMI |---Gomory's mixed integer cut;\\
1.558 +&\verb|GLP_RF_MIR |---mixed integer rounding cut;\\
1.559 +&\verb|GLP_RF_COV |---mixed cover cut;\\
1.560 +&\verb|GLP_RF_CLQ |---clique cut.\\
1.561 +\end{tabular}
1.562 +
1.563 +\newpage
1.564 +
1.565 +\subsection{glp\_ios\_mip\_gap---compute relative MIP gap}
1.566 +
1.567 +\subsubsection*{Synopsis}
1.568 +
1.569 +\begin{verbatim}
1.570 +double glp_ios_mip_gap(glp_tree *tree);
1.571 +\end{verbatim}
1.572 +
1.573 +\subsubsection*{Description}
1.574 +
1.575 +The routine \verb|glp_ios_mip_gap| computes the relative MIP gap (also
1.576 +called {\it duality gap}) with the following formula:
1.577 +$${\tt gap} = \frac{|{\tt best\_mip} - {\tt best\_bnd}|}
1.578 +{|{\tt best\_mip}| + {\tt DBL\_EPSILON}}$$
1.579 +where \verb|best_mip| is the best integer feasible solution found so
1.580 +far, \verb|best_bnd| is the best (global) bound. If no integer feasible
1.581 +solution has been found yet, \verb|gap| is set to \verb|DBL_MAX|.
1.582 +
1.583 +\subsubsection*{Returns}
1.584 +
1.585 +The routine \verb|glp_ios_mip_gap| returns the relative MIP gap.
1.586 +
1.587 +\subsubsection*{Comments}
1.588 +
1.589 +The relative MIP gap is used to measure the quality of the best integer
1.590 +feasible solution found so far, because the optimal solution value
1.591 +$z^*$ for the original MIP problem always lies in the range
1.592 +$${\tt best\_bnd}\leq z^*\leq{\tt best\_mip}$$
1.593 +in case of minimization, or in the range
1.594 +$${\tt best\_mip}\leq z^*\leq{\tt best\_bnd}$$
1.595 +in case of maximization.
1.596 +
1.597 +To express the relative MIP gap in percents the value returned by the
1.598 +routine \verb|glp_ios_mip_gap| should be multiplied by 100\%.
1.599 +
1.600 +\newpage
1.601 +
1.602 +\subsection{glp\_ios\_node\_data---access application-specific data}
1.603 +
1.604 +\subsubsection*{Synopsis}
1.605 +
1.606 +\begin{verbatim}
1.607 +void *glp_ios_node_data(glp_tree *tree, int p);
1.608 +\end{verbatim}
1.609 +
1.610 +\subsubsection*{Description}
1.611 +
1.612 +The routine \verb|glp_ios_node_data| allows the application accessing a
1.613 +memory block allocated for the subproblem (which may be active or
1.614 +inactive), whose reference number is $p$.
1.615 +
1.616 +The size of the block is defined by the control parameter \verb|cb_size|
1.617 +passed to the routine \verb|glp_intopt|. The block is initialized by
1.618 +binary zeros on creating corresponding subproblem, and its contents is
1.619 +kept until the subproblem will be removed from the tree.
1.620 +
1.621 +The application may use these memory blocks to store specific data for
1.622 +each subproblem.
1.623 +
1.624 +\subsubsection*{Returns}
1.625 +
1.626 +The routine \verb|glp_ios_node_data| returns a pointer to the memory
1.627 +block for the specified subproblem. Note that if \verb|cb_size| = 0, the
1.628 +routine returns a null pointer.
1.629 +
1.630 +\subsection{glp\_ios\_select\_node---select subproblem to continue the
1.631 +search}
1.632 +
1.633 +\subsubsection*{Synopsis}
1.634 +
1.635 +\begin{verbatim}
1.636 +void glp_ios_select_node(glp_tree *tree, int p);
1.637 +\end{verbatim}
1.638 +
1.639 +\subsubsection*{Description}
1.640 +
1.641 +The routine \verb|glp_ios_select_node| can be called from the
1.642 +user-defined callback routine in response to the reason
1.643 +\verb|GLP_ISELECT| to select an active subproblem, whose reference
1.644 +number is $p$. The search will be continued from the subproblem
1.645 +selected.
1.646 +
1.647 +\newpage
1.648 +
1.649 +\subsection{glp\_ios\_heur\_sol---provide solution found by heuristic}
1.650 +
1.651 +\subsubsection*{Synopsis}
1.652 +
1.653 +\begin{verbatim}
1.654 +int glp_ios_heur_sol(glp_tree *tree, const double x[]);
1.655 +\end{verbatim}
1.656 +
1.657 +\subsubsection*{Description}
1.658 +
1.659 +The routine \verb|glp_ios_heur_sol| can be called from the user-defined
1.660 +callback routine in response to the reason \verb|GLP_IHEUR| to provide
1.661 +an integer feasible solution found by a primal heuristic.
1.662 +
1.663 +Primal values of {\it all} variables (columns) found by the heuristic
1.664 +should be placed in locations $x[1]$, \dots, $x[n]$, where $n$ is the
1.665 +number of columns in the original problem object. Note that the routine
1.666 +\verb|glp_ios_heur_sol| does {\it not} check primal feasibility of the
1.667 +solution provided.
1.668 +
1.669 +Using the solution passed in the array $x$ the routine computes value
1.670 +of the objective function. If the objective value is better than the
1.671 +best known integer feasible solution, the routine computes values of
1.672 +auxiliary variables (rows) and stores all solution components in the
1.673 +problem object.
1.674 +
1.675 +\subsubsection*{Returns}
1.676 +
1.677 +If the provided solution is accepted, the routine
1.678 +\verb|glp_ios_heur_sol| returns zero. Otherwise, if the provided
1.679 +solution is rejected, the routine returns non-zero.
1.680 +
1.681 +\subsection{glp\_ios\_can\_branch---check if can branch upon specified
1.682 +variable}
1.683 +
1.684 +\subsubsection*{Synopsis}
1.685 +
1.686 +\begin{verbatim}
1.687 +int glp_ios_can_branch(glp_tree *tree, int j);
1.688 +\end{verbatim}
1.689 +
1.690 +\subsubsection*{Returns}
1.691 +
1.692 +If $j$-th variable (column) can be used to branch upon, the routine
1.693 +returns non-zero, otherwise zero.
1.694 +
1.695 +\newpage
1.696 +
1.697 +\subsection{glp\_ios\_branch\_upon---choose variable to branch upon}
1.698 +
1.699 +\subsubsection*{Synopsis}
1.700 +
1.701 +\begin{verbatim}
1.702 +void glp_ios_branch_upon(glp_tree *tree, int j, int sel);
1.703 +\end{verbatim}
1.704 +
1.705 +\subsubsection*{Description}
1.706 +
1.707 +The routine \verb|glp_ios_branch_upon| can be called from the
1.708 +user-defined callback routine in response to the reason
1.709 +\verb|GLP_IBRANCH| to choose a branching variable, whose ordinal number
1.710 +is $j$. Should note that only variables, for which the routine
1.711 +\verb|glp_ios_can_branch| returns non-zero, can be used to branch upon.
1.712 +
1.713 +The parameter \verb|sel| is a flag that indicates which branch
1.714 +(subproblem) should be selected next to continue the search:
1.715 +
1.716 +\verb|GLP_DN_BRNCH|---select down-branch;
1.717 +
1.718 +\verb|GLP_UP_BRNCH|---select up-branch;
1.719 +
1.720 +\verb|GLP_NO_BRNCH|---use general selection technique.
1.721 +
1.722 +\subsubsection*{Comments}
1.723 +
1.724 +On branching the solver removes the current active subproblem from the
1.725 +active list and creates two new subproblems ({\it down-} and {\it
1.726 +up-branches}), which are added to the end of the active list. Note that
1.727 +the down-branch is created before the up-branch, so the last active
1.728 +subproblem will be the up-branch.
1.729 +
1.730 +The down- and up-branches are identical to the current subproblem with
1.731 +exception that in the down-branch the upper bound of $x_j$, the variable
1.732 +chosen to branch upon, is replaced by $\lfloor x_j^*\rfloor$, while in
1.733 +the up-branch the lower bound of $x_j$ is replaced by
1.734 +$\lceil x_j^*\rceil$, where $x_j^*$ is the value of $x_j$ in optimal
1.735 +solution to LP relaxation of the current subproblem. For example, if
1.736 +$x_j^*=3.14$, the new upper bound of $x_j$ in the down-branch is
1.737 +$\lfloor 3.14\rfloor=3$, and the new lower bound in the up-branch is
1.738 +$\lceil 3.14\rceil=4$.)
1.739 +
1.740 +Additionally the callback routine may select either down- or up-branch,
1.741 +from which the solver will continue the search. If none of the branches
1.742 +is selected, a general selection technique will be used.
1.743 +
1.744 +\newpage
1.745 +
1.746 +\subsection{glp\_ios\_terminate---terminate the solution process}
1.747 +
1.748 +\subsubsection*{Synopsis}
1.749 +
1.750 +\begin{verbatim}
1.751 +void glp_ios_terminate(glp_tree *tree);
1.752 +\end{verbatim}
1.753 +
1.754 +\subsubsection*{Description}
1.755 +
1.756 +The routine \verb|glp_ios_terminate| sets a flag indicating that the
1.757 +MIP solver should prematurely terminate the search.
1.758 +
1.759 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1.760 +
1.761 +\newpage
1.762 +
1.763 +\section{The search tree exploring routines}
1.764 +
1.765 +\subsection{glp\_ios\_tree\_size---determine size of the search tree}
1.766 +
1.767 +\subsubsection*{Synopsis}
1.768 +
1.769 +\begin{verbatim}
1.770 +void glp_ios_tree_size(glp_tree *tree, int *a_cnt, int *n_cnt,
1.771 + int *t_cnt);
1.772 +\end{verbatim}
1.773 +
1.774 +\subsubsection*{Description}
1.775 +
1.776 +The routine \verb|glp_ios_tree_size| stores the following three counts
1.777 +which characterize the current size of the search tree:
1.778 +
1.779 +\verb|a_cnt| is the current number of active nodes, i.e. the current
1.780 +size of the active list;
1.781 +
1.782 +\verb|n_cnt| is the current number of all (active and inactive) nodes;
1.783 +
1.784 +\verb|t_cnt| is the total number of nodes including those which have
1.785 +been already removed from the tree. This count is increased whenever
1.786 +a new node appears in the tree and never decreased.
1.787 +
1.788 +If some of the parameters \verb|a_cnt|, \verb|n_cnt|, \verb|t_cnt| is
1.789 +a null pointer, the corresponding count is not stored.
1.790 +
1.791 +\subsection{glp\_ios\_curr\_node---determine current active subprob-\\
1.792 +lem}
1.793 +
1.794 +\subsubsection*{Synopsis}
1.795 +
1.796 +\begin{verbatim}
1.797 +int glp_ios_curr_node(glp_tree *tree);
1.798 +\end{verbatim}
1.799 +
1.800 +\subsubsection*{Returns}
1.801 +
1.802 +The routine \verb|glp_ios_curr_node| returns the reference number of the
1.803 +current active subproblem. However, if the current subproblem does not
1.804 +exist, the routine returns zero.
1.805 +
1.806 +\newpage
1.807 +
1.808 +\subsection{glp\_ios\_next\_node---determine next active subproblem}
1.809 +
1.810 +\subsubsection*{Synopsis}
1.811 +
1.812 +\begin{verbatim}
1.813 +int glp_ios_next_node(glp_tree *tree, int p);
1.814 +\end{verbatim}
1.815 +
1.816 +\subsubsection*{Returns}
1.817 +
1.818 +If the parameter $p$ is zero, the routine \verb|glp_ios_next_node|
1.819 +returns the reference number of the first active subproblem. However,
1.820 +if the tree is empty, zero is returned.
1.821 +
1.822 +If the parameter $p$ is not zero, it must specify the reference number
1.823 +of some active subproblem, in which case the routine returns the
1.824 +reference number of the next active subproblem. However, if there is
1.825 +no next active subproblem in the list, zero is returned.
1.826 +
1.827 +All subproblems in the active list are ordered chronologically, i.e.
1.828 +subproblem $A$ precedes subproblem $B$ if $A$ was created before $B$.
1.829 +
1.830 +\subsection{glp\_ios\_prev\_node---determine previous active subproblem}
1.831 +
1.832 +\subsubsection*{Synopsis}
1.833 +
1.834 +\begin{verbatim}
1.835 +int glp_ios_prev_node(glp_tree *tree, int p);
1.836 +\end{verbatim}
1.837 +
1.838 +\subsubsection*{Returns}
1.839 +
1.840 +If the parameter $p$ is zero, the routine \verb|glp_ios_prev_node|
1.841 +returns the reference number of the last active subproblem. However, if
1.842 +the tree is empty, zero is returned.
1.843 +
1.844 +If the parameter $p$ is not zero, it must specify the reference number
1.845 +of some active subproblem, in which case the routine returns the
1.846 +reference number of the previous active subproblem. However, if there
1.847 +is no previous active subproblem in the list, zero is returned.
1.848 +
1.849 +All subproblems in the active list are ordered chronologically, i.e.
1.850 +subproblem $A$ precedes subproblem $B$ if $A$ was created before $B$.
1.851 +
1.852 +\newpage
1.853 +
1.854 +\subsection{glp\_ios\_up\_node---determine parent subproblem}
1.855 +
1.856 +\subsubsection*{Synopsis}
1.857 +
1.858 +\begin{verbatim}
1.859 +int glp_ios_up_node(glp_tree *tree, int p);
1.860 +\end{verbatim}
1.861 +
1.862 +\subsubsection*{Returns}
1.863 +
1.864 +The parameter $p$ must specify the reference number of some (active or
1.865 +inactive) subproblem, in which case the routine \verb|iet_get_up_node|
1.866 +returns the reference number of its parent subproblem. However, if the
1.867 +specified subproblem is the root of the tree and, therefore, has
1.868 +no parent, the routine returns zero.
1.869 +
1.870 +\subsection{glp\_ios\_node\_level---determine subproblem level}
1.871 +
1.872 +\subsubsection*{Synopsis}
1.873 +
1.874 +\begin{verbatim}
1.875 +int glp_ios_node_level(glp_tree *tree, int p);
1.876 +\end{verbatim}
1.877 +
1.878 +\subsubsection*{Returns}
1.879 +
1.880 +The routine \verb|glp_ios_node_level| returns the level of the
1.881 +subproblem,\linebreak whose reference number is $p$, in the
1.882 +branch-and-bound tree. (The root subproblem has level 0, and the level
1.883 +of any other subproblem is the level of its parent plus one.)
1.884 +
1.885 +\subsection{glp\_ios\_node\_bound---determine subproblem local\\bound}
1.886 +
1.887 +\subsubsection*{Synopsis}
1.888 +
1.889 +\begin{verbatim}
1.890 +double glp_ios_node_bound(glp_tree *tree, int p);
1.891 +\end{verbatim}
1.892 +
1.893 +\subsubsection*{Returns}
1.894 +
1.895 +The routine \verb|glp_ios_node_bound| returns the local bound for
1.896 +(active or inactive) subproblem, whose reference number is $p$.
1.897 +
1.898 +\subsubsection*{Comments}
1.899 +
1.900 +The local bound for subproblem $p$ is an lower (minimization) or upper
1.901 +(maximization) bound for integer optimal solution to {\it this}
1.902 +subproblem (not to the original problem). This bound is local in the
1.903 +sense that only subproblems in the subtree rooted at node $p$ cannot
1.904 +have better integer feasible solutions.
1.905 +
1.906 +On creating a subproblem (due to the branching step) its local bound is
1.907 +inherited from its parent and then may get only stronger (never weaker).
1.908 +For the root subproblem its local bound is initially set to
1.909 +\verb|-DBL_MAX| (minimization) or \verb|+DBL_MAX| (maximization) and
1.910 +then improved as the root LP relaxation has been solved.
1.911 +
1.912 +Note that the local bound is not necessarily the optimal objective value
1.913 +to corresponding LP relaxation.
1.914 +
1.915 +\subsection{glp\_ios\_best\_node---find active subproblem with best
1.916 +local bound}
1.917 +
1.918 +\subsubsection*{Synopsis}
1.919 +
1.920 +\begin{verbatim}
1.921 +int glp_ios_best_node(glp_tree *tree);
1.922 +\end{verbatim}
1.923 +
1.924 +\subsubsection*{Returns}
1.925 +
1.926 +The routine \verb|glp_ios_best_node| returns the reference number of
1.927 +the active subproblem, whose local bound is best (i.e. smallest in case
1.928 +of minimization or largest in case of maximization). However, if the
1.929 +tree is empty, the routine returns zero.
1.930 +
1.931 +\subsubsection*{Comments}
1.932 +
1.933 +The best local bound is an lower (minimization) or upper (maximization)
1.934 +bound for integer optimal solution to the original MIP problem.
1.935 +
1.936 +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1.937 +
1.938 +\newpage
1.939 +
1.940 +\section{The cut pool routines}
1.941 +
1.942 +\subsection{glp\_ios\_pool\_size---determine current size of the cut\\
1.943 +pool}
1.944 +
1.945 +\subsubsection*{Synopsis}
1.946 +
1.947 +\begin{verbatim}
1.948 +int glp_ios_pool_size(glp_tree *tree);
1.949 +\end{verbatim}
1.950 +
1.951 +\subsubsection*{Returns}
1.952 +
1.953 +The routine \verb|glp_ios_pool_size| returns the current size of the
1.954 +cut pool, that is, the number of cutting plane constraints currently
1.955 +added to it.
1.956 +
1.957 +\subsection{glp\_ios\_add\_row---add constraint to the cut pool}
1.958 +
1.959 +\subsubsection*{Synopsis}
1.960 +
1.961 +\begin{verbatim}
1.962 +int glp_ios_add_row(glp_tree *tree, const char *name,
1.963 + int klass, int flags, int len, const int ind[],
1.964 + const double val[], int type, double rhs);
1.965 +\end{verbatim}
1.966 +
1.967 +\subsubsection*{Description}
1.968 +
1.969 +The routine \verb|glp_ios_add_row| adds specified row (cutting plane
1.970 +constraint) to the cut pool.
1.971 +
1.972 +The cutting plane constraint should have the following format:
1.973 +$$\sum_{j\in J}a_jx_j\left\{\begin{array}{@{}c@{}}\geq\\\leq\\
1.974 +\end{array}\right\}b,$$
1.975 +where $J$ is a set of indices (ordinal numbers) of structural variables,
1.976 +$a_j$ are constraint coefficients, $x_j$ are structural variables, $b$
1.977 +is the right-hand side.
1.978 +
1.979 +The parameter \verb|name| specifies a symbolic name assigned to the
1.980 +constraint (1 up to 255 characters). If it is \verb|NULL| or an empty
1.981 +string, no name is assigned.
1.982 +
1.983 +The parameter \verb|klass| specifies the constraint class, which must
1.984 +be either zero or a number in the range from 101 to 200. The application
1.985 +may use this attribute to distinguish between cutting plane constraints
1.986 +of different classes.\footnote{Constraint classes numbered from 1 to 100
1.987 +are reserved for GLPK cutting plane generators.}
1.988 +
1.989 +The parameter \verb|flags| currently is not used and must be zero.
1.990 +
1.991 +Ordinal numbers of structural variables (i.e. column indices) $j\in J$
1.992 +and numerical values of corresponding constraint coefficients $a_j$ must
1.993 +be placed in locations \verb|ind[1]|, \dots, \verb|ind[len]| and
1.994 +\verb|val[1]|, \dots, \verb|val[len]|, respectively, where
1.995 +${\tt len}=|J|$ is the number of constraint coefficients,
1.996 +$0\leq{\tt len}\leq n$, and $n$ is the number of columns in the problem
1.997 +object. Coefficients with identical column indices are not allowed.
1.998 +Zero coefficients are allowed, however, they are ignored.
1.999 +
1.1000 +The parameter \verb|type| specifies the constraint type as follows:
1.1001 +
1.1002 +\verb|GLP_LO| means inequality constraint $\Sigma a_jx_j\geq b$;
1.1003 +
1.1004 +\verb|GLP_UP| means inequality constraint $\Sigma a_jx_j\leq b$;
1.1005 +
1.1006 +The parameter \verb|rhs| specifies the right-hand side $b$.
1.1007 +
1.1008 +All cutting plane constraints in the cut pool are identified by their
1.1009 +ordinal numbers 1, 2, \dots, $size$, where $size$ is the current size
1.1010 +of the cut pool. New constraints are always added to the end of the cut
1.1011 +pool, thus, ordinal numbers of previously added constraints are not
1.1012 +changed.
1.1013 +
1.1014 +\subsubsection*{Returns}
1.1015 +
1.1016 +The routine \verb|glp_ios_add_row| returns the ordinal number of the
1.1017 +cutting plane constraint added, which is the new size of the cut pool.
1.1018 +
1.1019 +\subsubsection*{Example}
1.1020 +
1.1021 +\begin{verbatim}
1.1022 +/* generate triangle cutting plane:
1.1023 + x[i] + x[j] + x[k] <= 1 */
1.1024 +. . .
1.1025 +/* add the constraint to the cut pool */
1.1026 +ind[1] = i, val[1] = 1.0;
1.1027 +ind[2] = j, val[2] = 1.0;
1.1028 +ind[3] = k, val[3] = 1.0;
1.1029 +glp_ios_add_row(tree, NULL, TRIANGLE_CUT, 0, 3, ind, val,
1.1030 + GLP_UP, 1.0);
1.1031 +\end{verbatim}
1.1032 +
1.1033 +\subsubsection*{Comments}
1.1034 +
1.1035 +Cutting plane constraints added to the cut pool are intended to be then
1.1036 +added only to the {\it current} subproblem, so these constraints can be
1.1037 +globally as well as locally valid. However, adding a constraint to the
1.1038 +cut pool does not mean that it will be added to the current
1.1039 +subproblem---it depends on the solver's decision: if the constraint
1.1040 +seems to be efficient, it is moved from the pool to the current
1.1041 +subproblem, otherwise it is simply dropped.\footnote{Globally valid
1.1042 +constraints could be saved and then re-used for other subproblems, but
1.1043 +currently such feature is not implemented.}
1.1044 +
1.1045 +Normally, every time the callback routine is called for cut generation,
1.1046 +the cut pool is empty. On the other hand, the solver itself can generate
1.1047 +cutting plane constraints (like Gomory's or mixed integer rounding
1.1048 +cuts), in which case the cut pool may be non-empty.
1.1049 +
1.1050 +\subsection{glp\_ios\_del\_row---remove constraint from the cut pool}
1.1051 +
1.1052 +\subsubsection*{Synopsis}
1.1053 +
1.1054 +\begin{verbatim}
1.1055 +void glp_ios_del_row(glp_tree *tree, int i);
1.1056 +\end{verbatim}
1.1057 +
1.1058 +\subsubsection*{Description}
1.1059 +
1.1060 +The routine \verb|glp_ios_del_row| deletes $i$-th row (cutting plane
1.1061 +constraint) from the cut pool, where $1\leq i\leq size$ is the ordinal
1.1062 +number of the constraint in the pool, $size$ is the current size of the
1.1063 +cut pool.
1.1064 +
1.1065 +Note that deleting a constraint from the cut pool leads to changing
1.1066 +ordinal numbers of other constraints remaining in the pool. New ordinal
1.1067 +numbers of the remaining constraints are assigned under assumption that
1.1068 +the original order of constraints is not changed. Let, for example,
1.1069 +there be four constraints $a$, $b$, $c$ and $d$ in the cut pool, which
1.1070 +have ordinal numbers 1, 2, 3 and 4, respectively, and let constraint
1.1071 +$b$ have been deleted. Then after deletion the remaining constraint $a$,
1.1072 +$c$ and $d$ are assigned new ordinal numbers 1, 2 and 3, respectively.
1.1073 +
1.1074 +To find the constraint to be deleted the routine \verb|glp_ios_del_row|
1.1075 +uses ``smart'' linear search, so it is recommended to remove constraints
1.1076 +in a natural or reverse order and avoid removing them in a random order.
1.1077 +
1.1078 +\subsubsection*{Example}
1.1079 +
1.1080 +\begin{verbatim}
1.1081 +/* keep first 10 constraints in the cut pool and remove other
1.1082 + constraints */
1.1083 +while (glp_ios_pool_size(tree) > 10)
1.1084 + glp_ios_del_row(tree, glp_ios_pool_size(tree));
1.1085 +\end{verbatim}
1.1086 +
1.1087 +\newpage
1.1088 +
1.1089 +\subsection{glp\_ios\_clear\_pool---remove all constraints from the cut
1.1090 +pool}
1.1091 +
1.1092 +\subsubsection*{Synopsis}
1.1093 +
1.1094 +\begin{verbatim}
1.1095 +void glp_ios_clear_pool(glp_tree *tree);
1.1096 +\end{verbatim}
1.1097 +
1.1098 +\subsubsection*{Description}
1.1099 +
1.1100 +The routine \verb|glp_ios_clear_pool| makes the cut pool empty deleting
1.1101 +all existing rows (cutting plane constraints) from it.
1.1102 +
1.1103 +%* eof *%