# Selecting the Next Node in the Search Tree

## CbcCompare - Comparison Methods

The order in which the nodes of the search tree are explored can strongly influence the performance of branch-and-cut algorithms. CBC give users complete control over the search order. The search order is controlled via the CbcCompare... class. CBC provides an abstract base class, CbcCompareBase, and several commonly used instances which are described in .

Compare Classes Provided

Class name | Description
--- | ---
CbcCompareDepth | This will always choose the node deepest in tree. It gives minimum tree size but may take a long time to find the best solution.
CbcCompareObjective | This will always choose the node with the best objective value. This may give a very large tree. It is likely that the first solution found will be the best and the search should finish soon after the first solution is found.
CbcCompareDefault | This is designed to do a mostly depth-first search until a solution has been found. It then use estimates that are designed to give a slightly better solution. If a reasonable number of nodes have been explored (or a reasonable number of solutions found), then this class will adopt a breadth-first search (i.e., making a comparison based strictly on objective function values) unless the tree is very large when it will revert to depth-first search. A better description of CbcCompareUser is given below.
CbcCompareEstimate | When pseudo costs are invoked, they can be used to guess a solution. This class uses the guessed solution.

It is relatively simple for an experienced user to create new compare class instances. The code in describes how to build a new comparison class and the reasoning behind it. The complete source can be found in `CbcCompareUser.hpp` and `CbcCompareUser.cpp`, located in the CBC Samples directory. See . The key method in CbcCompare is `bool test(CbcNode* x, CbcNode* y))` which returns true if node y is preferred over node x. In the `test()` method, information from CbcNode can easily be used. list some commonly used methods to access information at a node.

Information Available from CbcNode

`double objectiveValue() const` | Value of objective at the node.
`int numberUnsatisfied() const` | Number of unsatisfied integers (assuming branching object is an integer - otherwise it might be number of unsatisfied sets).
`int depth() const` | Depth of the node in the search tree.
`double guessedObjectiveValue() const` | If user was setting this (e.g., if using pseudo costs).
`int way() const` | The way which branching would next occur from this node (for more advanced use).
`int variable() const` | The branching "variable" (associated with the CbcBranchingObject -- for more advanced use).

The node desired in the tree is often a function of the how the search is progressing. In the design of CBC, there is no information on the state of the tree. The CBC is designed so that the method `newSolution()` is called whenever a solution is found and the method `every1000Nodes()` is called every 1000 nodes. When these methods are called, the user has the opportunity to modify the behavior of `test()` by adjusting their common variables (e.g., `weight_`). Because CbcNode has a pointer to the model, the user can also influence the search through actions such as changing the maximum time CBC is allowed, once a solution has been found (e.g., `CbcModel::setMaximumSeconds(double value)`). In `CbcCompareUser.cpp` of the `COIN/Cbc/Samples` directory, four items of data are used.

- 1) The number of solutions found so far

- 2) The size of the tree (defined to be the number of active nodes)

- 3) A weight, `weight_`, which is initialized to -1.0

- 4) A saved value of weight, `saveWeight_` (for when weight is set back to -1.0 for special reason)

The full code for the `CbcCompareUser::test()` method is given in .

`CbcCompareUser::test()`

    // Returns true if y better than x
    bool
    CbcCompareUser::test (CbcNode * x, CbcNode * y)
    {
      if (weight_==-1.0) {
        // before solution
        if (x->numberUnsatisfied() > y->numberUnsatisfied())
          return true;
        else if (x->numberUnsatisfied() < y->numberUnsatisfied())
          return false;
        else
          return x->depth() < y->depth();
      } else {
        // after solution.
        // note: if weight_=0, comparison is based
        //       solely on objective value
        double weight = CoinMax(weight_,0.0);
        return x->objectiveValue()+ weight*x->numberUnsatisfied() >
          y->objectiveValue() + weight*y->numberUnsatisfied();
      }
    }

Initially, `weight`_ is -1.0 and the search is biased towards depth first. In fact, `test()` prefers y if y has fewer unsatisfied variables. In the case of a tie, `test()` prefers the node with the greater depth in tree. Once a solution is found, `newSolution()` is called. The method `newSolution()` interacts with `test()` by means of the variable `weight_`. If the solution was achieved by branching, a calculation is made to determine the cost per unsatisfied integer variable to go from the continuous solution to an integer solution. The variable `weight_` is then set to aim at a slightly better solution. From then on, `test()` returns true if it seems that y will lead to a better solution than x. This source for `newSolution()` in given in .

`CbcCompareUser::newSolution()`

    // This allows the test() method to change behavior by resetting weight_.
    // It is called after each new solution is found.
    void
    CbcCompareUser::newSolution(CbcModel * model,
    			       double objectiveAtContinuous,
    			       int numberInfeasibilitiesAtContinuous)
    {
      if (model->getSolutionCount()==model->getNumberHeuristicSolutions())
        return; // solution was found by rounding so ignore it.

      // set weight_ to get close to this solution
      double costPerInteger =
        (model->getObjValue()-objectiveAtContinuous)/
        ((double) numberInfeasibilitiesAtContinuous);
      weight_ = 0.98*costPerInteger;
      saveWeight_=weight_;
      numberSolutions_++;
      if (numberSolutions_>5)
        weight_ =0.0; // comparison in test() will be
                      // based strictly on objective value.
    }

As the search progresses, the comparison can be modified. If many nodes (or many solutions) have been genereated, then `weight_` is set to 0.0 leading to a breadth-first search. Breadth-first search can lead to an enormous tree. If the tree size is exceeds 10000, it may be desirable to return to a search biased towards depth first. Changing the behavior in this manner is done by the method `every1000Nodes` shown in .

`CbcCompareUser::every1000Nodes()`

    // This allows the test() method to change behavior every so often
    bool
    CbcCompareUser::every1000Nodes(CbcModel * model, int numberNodes)
    {
      if (numberNodes>10000)
        weight_ =0.0; // compare nodes based on objective value
      else if (numberNodes==1000&&weight_==-2.0)
        weight_=-1.0; // Go to depth first
      // get size of tree
      treeSize_ = model->tree()->size();
      if (treeSize_>10000) {
        // set weight to reduce size most of time
        if (treeSize_>20000)
          weight_=-1.0;
        else if ((numberNodes%4000)!=0)
          weight_=-1.0;
        else
          weight_=saveWeight_;
      }
      return numberNodes==11000; // resort if first time
    }

# Getting Good Bounds in CBC

## CbcHeuristic - Heuristic Methods

In practice, it is very useful to get a good solution reasonably fast. Any MIP-feasible solution produces an upper bound, and a good bound will greatly reduce the run time. Good solutions can satisfy the user on very large problems where a complete search is impossible. Obviously, heuristics are problem dependent, although some do have more general use. At present there is only one heuristic in CBC itself, CbcRounding. Hopefully, the number will grow. Other heuristics are in the `COIN/Cbc/Samples` directory. A heuristic tries to obtain a solution to the original problem so it only needs to consider the original rows and does not have to use the current bounds. CBC provides an abstract base class CbcHeuristic and a rounding heuristic in CBC.

This chapter describes how to build a greedy heuristic for a set covering problem, e.g., the miplib problem fast0507. A more general (and efficient) version of the heuristic is in `CbcHeuristicGreedy.hpp` and `CbcHeuristicGreedy.cpp` located in the `COIN/Cbc/Samples` directory, see .

The greedy heuristic will leave all variables taking value one at this node of the tree at value one, and will initially set all other variable to value zero. All variables are then sorted in order of their cost divided by the number of entries in rows which are not yet covered. (We may randomize that value a bit so that ties will be broken in different ways on different runs of the heuristic.) The best one is choosen, and set to one. The process is repeated. Because this is a set covering problem (i.e., all constraints are ≥), the heuristic is guaranteed to find a solution (but not necessarily an improved solution). The speed of the heuristic could be improved by just redoing those affected, but for illustrative purposes we will keep it simple.(The speed could also be improved if all elements are 1.0).

The key CbcHeuristic method is `int solution(double & solutionValue, double * betterSolution)`. The `solution()` method returns 0 if no solution found, and returns 1 if a solution is found, in which case it fills in the objective value and primal solution. The code in `CbcHeuristicGreedy.cpp` is a little more complicated than this following example. For instance, the code here assumes all variables are integer. The important bit of data is a copy of the matrix (stored by column) before any cuts have been made. The data used are bounds, objective and the matrix plus two work arrays.

Data

      OsiSolverInterface * solver = model_->solver(); // Get solver from CbcModel
      const double * columnLower = solver->getColLower(); // Column Bounds
      const double * columnUpper = solver->getColUpper();
      const double * rowLower = solver->getRowLower(); // We know we only need lower bounds
      const double * solution = solver->getColSolution();
      const double * objective = solver->getObjCoefficients(); // In code we also use min/max
      double integerTolerance = model_->getDblParam(CbcModel::CbcIntegerTolerance);
      double primalTolerance;
      solver->getDblParam(OsiPrimalTolerance,primalTolerance);
      int numberRows = originalNumberRows_; // This is number of rows when matrix was passed in
      // Column copy of matrix (before cuts)
      const double * element = matrix_.getElements();
      const int * row = matrix_.getIndices();
      const CoinBigIndex * columnStart = matrix_.getVectorStarts();
      const int * columnLength = matrix_.getVectorLengths();

      // Get solution array for heuristic solution
      int numberColumns = solver->getNumCols();
      double * newSolution = new double [numberColumns];
      // And to sum row activities
      double * rowActivity = new double[numberRows];

The `newSolution` is then initialized to the rounded down solution.

Initialize `newSolution`

      for (iColumn=0;iColumn<numberColumns;iColumn++) {
        CoinBigIndex j;
        double value = solution[iColumn];
        // Round down integer
        if (fabs(floor(value+0.5)-value)<integerTolerance)
          value=floor(CoinMax(value+1.0e-3,columnLower[iColumn]));
        // make sure clean
        value = CoinMin(value,columnUpper[iColumn]);
        value = CoinMax(value,columnLower[iColumn]);
        newSolution[iColumn]=value;
        if (value) {
          double cost = direction * objective[iColumn];
          newSolutionValue += value*cost;
          for (j=columnStart[iColumn];
               j<columnStart[iColumn]+columnLength[iColumn];j++) {
            int iRow=row[j];
            rowActivity[iRow] += value*element[j];
          }
        }
      }

At this point some row activities may be below their lower bound. To correct this infeasibility, the variable which is cheapest in reducing the sum of infeasibilities is found and updated, and the process repeats. This is a finite process. (Theimplementation could be faster, but is kept simple for illustrative purposes.)

Create Feasible `newSolution` from Initial `newSolution`

      while (true) {
        // Get column with best ratio
        int bestColumn=-1;
        double bestRatio=COIN_DBL_MAX;
        for (int iColumn=0;iColumn<numberColumns;iColumn++) {
          CoinBigIndex j;
          double value = newSolution[iColumn];
          double cost = direction * objective[iColumn];
          // we could use original upper rather than current
          if (value+0.99<columnUpper[iColumn]) {
            double sum=0.0; // Compute how much we will reduce infeasibility by
            for (j=columnStart[iColumn];
                 j<columnStart[iColumn]+columnLength[iColumn];j++) {
              int iRow=row[j];
              double gap = rowLower[iRow]-rowActivity[iRow];
              if (gap>1.0e-7) {
                sum += CoinMin(element[j],gap);
              if (element[j]+rowActivity[iRow]<rowLower[iRow]+1.0e-7) {
                sum += element[j];
    	  }
    	}
            if (sum>0.0) {
    	  double ratio = (cost/sum)*(1.0+0.1*CoinDrand48());
    	  if (ratio<bestRatio) {
                bestRatio=ratio;
                bestColumn=iColumn;
              }
    	}
          }
        }
        if (bestColumn<0)
          break; // we have finished
        // Increase chosen column
        newSolution[bestColumn] += 1.0;
        double cost = direction * objective[bestColumn];
        newSolutionValue += cost;
        for (CoinBigIndex j=columnStart[bestColumn];
             j<columnStart[bestColumn]+columnLength[bestColumn];j++) {
          int iRow = row[j];
          rowActivity[iRow] += element[j];
        }
      }

A solution value of `newSolution` is compared to the best solution value. If `newSolution` is an improvement, its feasibility is validated.

Check Solution Quality of `newSolution`

      returnCode=0; // 0 means no good solution
      if (newSolutionValue<solutionValue) { // minimization
        // check feasible
        memset(rowActivity,0,numberRows*sizeof(double));
        for (iColumn=0;iColumn<numberColumns;iColumn++) {
          CoinBigIndex j;
          double value = newSolution[iColumn];
          if (value) {
    	for (j=columnStart[iColumn];
    	     j<columnStart[iColumn]+columnLength[iColumn];j++) {
    	  int iRow=row[j];
    	  rowActivity[iRow] += value*element[j];
    	}
          }
        }
        // check was approximately feasible
        bool feasible=true;
        for (iRow=0;iRow<numberRows;iRow++) {
          if(rowActivity[iRow]<rowLower[iRow]) {
    	if (rowActivity[iRow]<rowLower[iRow]-10.0*primalTolerance)
    	  feasible = false;
          }
        }
        if (feasible) {
          // new solution
          memcpy(betterSolution,newSolution,numberColumns*sizeof(double));
          solutionValue = newSolutionValue;
          // We have good solution
          returnCode=1;
        }
      }

# Branching

## Pseudo Cost Branching

If the user declares variables as integer but does no more, then CBC will treat them as simple integer variables. In many cases the user would like to do some more fine tuning. This section shows how to create integer variables with pseudo costs. When pseudo costs are given then it is assumed that if a variable is at 1.3 then the cost of branching that variable down will be 0.3 times the down pseudo cost and the cost of branching up would be 0.7 times the up pseudo cost. Pseudo costs can be used both for branching and for choosing a node. The full code is in `longthin.cpp` located in the CBC Samples directory, see .

The idea is simple for set covering problems. Branching up gets us much closer to an integer solution so we will encourage that direction by branch up if variable value > 0.333333. The expected cost of going up obviously depends on the cost of the variable. The pseudo costs are choosen to reflect that fact.

CbcSimpleIntegerPseudoCosts

      int iColumn;
      int numberColumns = solver3->getNumCols();
      // do pseudo costs
      CbcObject ** objects = new CbcObject * [numberColumns];
      // Point to objective
      const double * objective = model.getObjCoefficients();
      int numberIntegers=0;
      for (iColumn=0;iColumn<numberColumns;iColumn++) {
        if (solver3->isInteger(iColumn)) {
          double cost = objective[iColumn];
          CbcSimpleIntegerPseudoCost * newObject =
            new CbcSimpleIntegerPseudoCost(&model,numberIntegers,iColumn,
                                           2.0*cost,cost);
          newObject->setMethod(3);
          objects[numberIntegers++]= newObject;
        }
      }
      // Now add in objects (they will replace simple integers)
      model.addObjects(numberIntegers,objects);
      for (iColumn=0;iColumn<numberIntegers;iColumn++)
        delete objects[iColumn];
      delete [] objects;

The code in also tries to give more importance to variables with more coefficients. Whether this sort of thing is worthwhile should be the subject of experimentation.

## Follow-On Branching

In crew scheduling, the problems are long and thin. A problem may have a few rows but many thousands of variables. Branching a variable to 1 is very powerful as it fixes many other variables to zero, but branching to zero is very weak as thousands of variables can increase from zero. In crew scheduling problems, each constraint is a flight leg, e.g., JFK airport to DFW airport. From DFW there may be several flights the crew could take next - suppose one flight is the 9:30 flight from DFW to LAX airport. A binary branch is that the crew arriving at DFW either take the 9:30 flight to LAX or they don't. This "follow-on" branching does not fix individual variables. Instead this branching divides all the variables with entries in the JFK-DFW constraint into two groups - those with entries in the DFW-LAX constraint and those without entries.

The full sample code for follow-on brancing is in `crew.cpp` located in the CBC Samples directory, see ). In this case, the simple integer variables are left which may be necessary if other sorts of constraints exist. Follow-on branching rules are to be considered first, so the priorities are set to indicated the follow-on rules take precedence. Priority 1 is the highest priority.

CbcFollowOn

      int iColumn;
      int numberColumns = solver3->getNumCols();
      /* We are going to add a single follow-on object but we
         want to give low priority to existing integers
         As the default priority is 1000 we don't actually need to give
         integer priorities but it is here to show how.
      */
      // Normal integer priorities
      int * priority = new int [numberColumns];
      int numberIntegers=0;
      for (iColumn=0;iColumn<numberColumns;iColumn++) {
        if (solver3->isInteger(iColumn)) {
          priority[numberIntegers++]= 100; // low priority
        }
      }
      /* Second parameter is set to true for objects,
         and false for integers. This indicates integers */
      model.passInPriorities(priority,false);
      delete [] priority;
      /* Add in objects before we can give them a priority.
         In this case just one object
         - but it shows the general method
      */
      CbcObject ** objects = new CbcObject * [1];
      objects[0]=new CbcFollowOn(&model);
      model.addObjects(1,objects);
      delete objects[0];
      delete [] objects;
      // High priority
      int followPriority=1;
      model.passInPriorities(&followPriority,true);

# Advance Solver Uses

## Creating a Solver via Inheritance

CBC uses a generic OsiSolverInterface and its `resolve` capability. This does not give much flexibility so advanced users can inherit from their interface of choice. This section illustrates how to implement such a solver for a long thin problem, e.g., fast0507 again. As with the other examples in the Guide, the sample code is not guaranteed to be the fastest way to solve the problem. The main purpose of the example is to illustrate techniques. The full source is in `CbcSolver2.hpp` and `CbcSolver2.cpp` located in the CBC Samples directory, see .

The method `initialSolve` is called a few times in CBC, and provides a convenient starting point. The `modelPtr_` derives from OsiClpSolverInterface.

`initialSolve()`

      // modelPtr_ is of type ClpSimplex *
      modelPtr_->setLogLevel(1); // switch on a bit of printout
      modelPtr_->scaling(0); // We don't want scaling for fast0507
      setBasis(basis_,modelPtr_); // Put basis into ClpSimplex
      // Do long thin by sprint
      ClpSolve options;
      options.setSolveType(ClpSolve::usePrimalorSprint);
      options.setPresolveType(ClpSolve::presolveOff);
      options.setSpecialOption(1,3,15); // Do 15 sprint iterations
      modelPtr_->initialSolve(options); // solve problem
      basis_ = getBasis(modelPtr_); // save basis
      modelPtr_->setLogLevel(0); // switch off printout

The `resolve()` method is more complicated than `initialSolve()`. The main pieces of data are a counter `count_` which is incremented each solve and an integer array `node_` which stores the last time a variable was active in a solution. For the first few times, the normal Dual Simplex is called and `node_` array is updated.

First Few Solves

      if (count_<10) {
        OsiClpSolverInterface::resolve(); // Normal resolve
        if (modelPtr_->status()==0) {
          count_++; // feasible - save any nonzero or basic
          const double * solution = modelPtr_->primalColumnSolution();
          for (int i=0;i<numberColumns;i++) {
    	if (solution[i]>1.0e-6||modelPtr_->getStatus(i)==ClpSimplex::basic) {
    	  node_[i]=CoinMax(count_,node_[i]);
    	  howMany_[i]++;
    	}
          }
        } else {
          printf("infeasible early on\n");
        }
      }

After the first few solves, only those variables which took part in a solution in the last so many solves are used. As fast0507 is a set covering problem, any rows which are already covered can be taken out.

Create Small Problem

        int * whichRow = new int[numberRows]; // Array to say which rows used
        int * whichColumn = new int [numberColumns]; // Array to say which columns used
        int i;
        const double * lower = modelPtr_->columnLower();
        const double * upper = modelPtr_->columnUpper();
        setBasis(basis_,modelPtr_); // Set basis
        int nNewCol=0; // Number of columns in small model
        // Column copy of matrix
        const double * element = modelPtr_->matrix()->getElements();
        const int * row = modelPtr_->matrix()->getIndices();
        const CoinBigIndex * columnStart = modelPtr_->matrix()->getVectorStarts();
        const int * columnLength = modelPtr_->matrix()->getVectorLengths();

        int * rowActivity = new int[numberRows]; // Number of columns with entries in each row
        memset(rowActivity,0,numberRows*sizeof(int));
        int * rowActivity2 = new int[numberRows]; // Lower bound on row activity for each row
        memset(rowActivity2,0,numberRows*sizeof(int));
        char * mark = (char *) modelPtr_->dualColumnSolution(); // Get some space to mark columns
        memset(mark,0,numberColumns);
        for (i=0;i<numberColumns;i++) {
          bool choose = (node_[i]>count_-memory_&&node_[i]>0); // Choose if used recently
          // Take if used recently or active in some sense
          if ((choose&&upper[i])
    	  ||(modelPtr_->getStatus(i)!=ClpSimplex::atLowerBound&&
                 modelPtr_->getStatus(i)!=ClpSimplex::isFixed)
    	  ||lower[i]>0.0) {
            mark[i]=1; // mark as used
    	whichColumn[nNewCol++]=i; // add to list
            CoinBigIndex j;
            double value = upper[i];
            if (value) {
              for (j=columnStart[i];
                   j<columnStart[i]+columnLength[i];j++) {
                int iRow=row[j];
                assert (element[j]==1.0);
                rowActivity[iRow] ++; // This variable can cover this row
              }
              if (lower[i]>0.0) {
                for (j=columnStart[i];
                     j<columnStart[i]+columnLength[i];j++) {
                  int iRow=row[j];
                  rowActivity2[iRow] ++; // This row redundant
                }
              }
            }
          }
        }
        int nOK=0; // Use to count rows which can be covered
        int nNewRow=0; // Use to make list of rows needed
        for (i=0;i<numberRows;i++) {
          if (rowActivity[i])
            nOK++;
          if (!rowActivity2[i])
            whichRow[nNewRow++]=i; // not satisfied
          else
            modelPtr_->setRowStatus(i,ClpSimplex::basic); // make slack basic
        }
        if (nOK<numberRows) {
          // The variables we have do not cover rows - see if we can find any that do
          for (i=0;i<numberColumns;i++) {
            if (!mark[i]&&upper[i]) {
              CoinBigIndex j;
              int good=0;
              for (j=columnStart[i];
                   j<columnStart[i]+columnLength[i];j++) {
                int iRow=row[j];
                if (!rowActivity[iRow]) {
                  rowActivity[iRow] ++;
                  good++;
                }
              }
              if (good) {
                nOK+=good; // This covers - put in list
                whichColumn[nNewCol++]=i;
              }
            }
          }
        }
        delete [] rowActivity;
        delete [] rowActivity2;
        if (nOK<numberRows) {
          // By inspection the problem is infeasible - no need to solve
          modelPtr_->setProblemStatus(1);
          delete [] whichRow;
          delete [] whichColumn;
          printf("infeasible by inspection\n");
          return;
        }
        // Now make up a small model with the right rows and columns
        ClpSimplex *  temp = new ClpSimplex(modelPtr_,nNewRow,whichRow,nNewCol,whichColumn);

If the variables cover the rows, then the problem is feasible (no cuts are being used). If the rows were equality constraints, then this might not be the case. More work would be needed. After the solution, the reduct costs are checked. If any reduced costs are negative, the code goes back to the full problem and cleans up with Primal Simplex.

Check Optimal Solution

        temp->setDualObjectiveLimit(1.0e50); // Switch off dual cutoff as problem is restricted
        temp->dual(); // solve
        double * solution = modelPtr_->primalColumnSolution(); // put back solution
        const double * solution2 = temp->primalColumnSolution();
        memset(solution,0,numberColumns*sizeof(double));
        for (i=0;i<nNewCol;i++) {
          int iColumn = whichColumn[i];
          solution[iColumn]=solution2[i];
          modelPtr_->setStatus(iColumn,temp->getStatus(i));
        }
        double * rowSolution = modelPtr_->primalRowSolution();
        const double * rowSolution2 = temp->primalRowSolution();
        double * dual = modelPtr_->dualRowSolution();
        const double * dual2 = temp->dualRowSolution();
        memset(dual,0,numberRows*sizeof(double));
        for (i=0;i<nNewRow;i++) {
          int iRow=whichRow[i];
          modelPtr_->setRowStatus(iRow,temp->getRowStatus(i));
          rowSolution[iRow]=rowSolution2[i];
          dual[iRow]=dual2[i];
        }
        // See if optimal
        double * dj = modelPtr_->dualColumnSolution();
        // get reduced cost for large problem
        // this assumes minimization
        memcpy(dj,modelPtr_->objective(),numberColumns*sizeof(double));
        modelPtr_->transposeTimes(-1.0,dual,dj);
        modelPtr_->setObjectiveValue(temp->objectiveValue());
        modelPtr_->setProblemStatus(0);
        int nBad=0;

        for (i=0;i<numberColumns;i++) {
          if (modelPtr_->getStatus(i)==ClpSimplex::atLowerBound
              &&upper[i]>lower[i]&&dj[i]<-1.0e-5)
            nBad++;
        }
        // If necessary clean up with primal (and save some statistics)
        if (nBad) {
          timesBad_++;
          modelPtr_->primal(1);
          iterationsBad_ += modelPtr_->numberIterations();
        }

The array `node_` is updated, as for the first few solves. To give some idea of the effect of this tactic, the problem fast0507 has 63,009 variables but the small problem never has more than 4,000 variables. In only about ten percent of solves was it necessary to resolve, and then the average number of iterations on full problem was less than 20.

## Quadratic MIP

To give another example - again only for illustrative purposes -- it is possible to do quadratic MIP with CBC. In this case, we make `resolve` the same as `initialSolve`. The full code is in `ClpQuadInterface.hpp` and `ClpQuadInterface.cpp` located in the CBC Samples directory, see ).

Solving a Quadratic MIP

      // save cutoff
      double cutoff = modelPtr_->dualObjectiveLimit();
      modelPtr_->setDualObjectiveLimit(1.0e50);
      modelPtr_->scaling(0);
      modelPtr_->setLogLevel(0);
      // solve with no objective to get feasible solution
      setBasis(basis_,modelPtr_);
      modelPtr_->dual();
      basis_ = getBasis(modelPtr_);
      modelPtr_->setDualObjectiveLimit(cutoff);
      if (modelPtr_->problemStatus())
        return; // problem was infeasible
      // Now pass in quadratic objective
      ClpObjective * saveObjective  = modelPtr_->objectiveAsObject();
      modelPtr_->setObjectivePointer(quadraticObjective_);
      modelPtr_->primal();
      modelPtr_->setDualObjectiveLimit(cutoff);
      if (modelPtr_->objectiveValue()>cutoff)
        modelPtr_->setProblemStatus(1);
      modelPtr_->setObjectivePointer(saveObjective);