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    *      http://www.apache.org/licenses/LICENSE-2.0
   *
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  package org.apache.commons.math3.optim.linear;
  
  import java.util.ArrayList;
  import java.util.List;
  import org.apache.commons.math3.exception.TooManyIterationsException;
  import org.apache.commons.math3.optim.PointValuePair;
  import org.apache.commons.math3.util.Precision;
  
  /**
   * Solves a linear problem using the "Two-Phase Simplex" method.
   * <p>
   * <b>Note:</b> Depending on the problem definition, the default convergence criteria
   * may be too strict, resulting in {@link NoFeasibleSolutionException} or
   * {@link TooManyIterationsException}. In such a case it is advised to adjust these
   * criteria with more appropriate values, e.g. relaxing the epsilon value.
   * <p>
   * Default convergence criteria:
   * <ul>
   *   <li>Algorithm convergence: 1e-6</li>
   *   <li>Floating-point comparisons: 10 ulp</li>
   *   <li>Cut-Off value: 1e-12</li>
   * </ul>
   * <p>
   * The cut-off value has been introduced to zero out very small numbers in the Simplex tableau,
   * as these may lead to numerical instabilities due to the nature of the Simplex algorithm
   * (the pivot element is used as a denominator). If the problem definition is very tight, the
   * default cut-off value may be too small, thus it is advised to increase it to a larger value,
   * in accordance with the chosen epsilon.
   * <p>
   * It may also be counter-productive to provide a too large value for {@link
   * org.apache.commons.math3.optim.MaxIter MaxIter} as parameter in the call of {@link
   * #optimize(org.apache.commons.math3.optim.OptimizationData...) optimize(OptimizationData...)},
   * as the {@link SimplexSolver} will use different strategies depending on the current iteration
   * count. After half of the allowed max iterations has already been reached, the strategy to select
   * pivot rows will change in order to break possible cycles due to degenerate problems.
   *
   * @version $Id: SimplexSolver.java 1462503 2013-03-29 15:48:27Z luc $
   * @since 2.0
   */
  public class SimplexSolver extends LinearOptimizer {
      /** Default amount of error to accept in floating point comparisons (as ulps). */
      static final int DEFAULT_ULPS = 10;
  
      /** Default cut-off value. */
      static final double DEFAULT_CUT_OFF = 1e-12;
  
      /** Default amount of error to accept for algorithm convergence. */
      private static final double DEFAULT_EPSILON = 1.0e-6;
  
      /** Amount of error to accept for algorithm convergence. */
      private final double epsilon;
  
      /** Amount of error to accept in floating point comparisons (as ulps). */
      private final int maxUlps;
  
      /**
       * Cut-off value for entries in the tableau: values smaller than the cut-off
       * are treated as zero to improve numerical stability.
       */
      private final double cutOff;
  
      /**
       * Builds a simplex solver with default settings.
       */
      public SimplexSolver() {
          this(DEFAULT_EPSILON, DEFAULT_ULPS, DEFAULT_CUT_OFF);
      }
  
      /**
       * Builds a simplex solver with a specified accepted amount of error.
       *
       * @param epsilon Amount of error to accept for algorithm convergence.
       */
      public SimplexSolver(final double epsilon) {
          this(epsilon, DEFAULT_ULPS, DEFAULT_CUT_OFF);
      }
  
      /**
       * Builds a simplex solver with a specified accepted amount of error.
       *
       * @param epsilon Amount of error to accept for algorithm convergence.
       * @param maxUlps Amount of error to accept in floating point comparisons.
       */
     public SimplexSolver(final double epsilon, final int maxUlps) {
         this(epsilon, maxUlps, DEFAULT_CUT_OFF);
     }
 
     /**
      * Builds a simplex solver with a specified accepted amount of error.
      *
      * @param epsilon Amount of error to accept for algorithm convergence.
      * @param maxUlps Amount of error to accept in floating point comparisons.
      * @param cutOff Values smaller than the cutOff are treated as zero.
      */
     public SimplexSolver(final double epsilon, final int maxUlps, final double cutOff) {
         this.epsilon = epsilon;
         this.maxUlps = maxUlps;
         this.cutOff = cutOff;
     }
 
     /**
      * Returns the column with the most negative coefficient in the objective function row.
      *
      * @param tableau Simple tableau for the problem.
      * @return the column with the most negative coefficient.
      */
     private Integer getPivotColumn(SimplexTableau tableau) {
         double minValue = 0;
         Integer minPos = null;
         for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getWidth() - 1; i++) {
             final double entry = tableau.getEntry(0, i);
             // check if the entry is strictly smaller than the current minimum
             // do not use a ulp/epsilon check
             if (entry < minValue) {
                 minValue = entry;
                 minPos = i;
             }
         }
         return minPos;
     }
 
     /**
      * Returns the row with the minimum ratio as given by the minimum ratio test (MRT).
      *
      * @param tableau Simple tableau for the problem.
      * @param col Column to test the ratio of (see {@link #getPivotColumn(SimplexTableau)}).
      * @return the row with the minimum ratio.
      */
     private Integer getPivotRow(SimplexTableau tableau, final int col) {
         // create a list of all the rows that tie for the lowest score in the minimum ratio test
         List<Integer> minRatioPositions = new ArrayList<Integer>();
         double minRatio = Double.MAX_VALUE;
         for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) {
             final double rhs = tableau.getEntry(i, tableau.getWidth() - 1);
             final double entry = tableau.getEntry(i, col);
 
             if (Precision.compareTo(entry, 0d, maxUlps) > 0) {
                 final double ratio = rhs / entry;
                 // check if the entry is strictly equal to the current min ratio
                 // do not use a ulp/epsilon check
                 final int cmp = Double.compare(ratio, minRatio);
                 if (cmp == 0) {
                     minRatioPositions.add(i);
                 } else if (cmp < 0) {
                     minRatio = ratio;
                     minRatioPositions = new ArrayList<Integer>();
                     minRatioPositions.add(i);
                 }
             }
         }
 
         if (minRatioPositions.size() == 0) {
             return null;
         } else if (minRatioPositions.size() > 1) {
             // there's a degeneracy as indicated by a tie in the minimum ratio test
 
             // 1. check if there's an artificial variable that can be forced out of the basis
             if (tableau.getNumArtificialVariables() > 0) {
                 for (Integer row : minRatioPositions) {
                     for (int i = 0; i < tableau.getNumArtificialVariables(); i++) {
                         int column = i + tableau.getArtificialVariableOffset();
                         final double entry = tableau.getEntry(row, column);
                         if (Precision.equals(entry, 1d, maxUlps) && row.equals(tableau.getBasicRow(column))) {
                             return row;
                         }
                     }
                 }
             }
 
             // 2. apply Bland's rule to prevent cycling:
             //    take the row for which the corresponding basic variable has the smallest index
             //
             // see http://www.stanford.edu/class/msande310/blandrule.pdf
             // see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper)
             //
             // Additional heuristic: if we did not get a solution after half of maxIterations
             //                       revert to the simple case of just returning the top-most row
             // This heuristic is based on empirical data gathered while investigating MATH-828.
             if (getEvaluations() < getMaxEvaluations() / 2) {
                 Integer minRow = null;
                 int minIndex = tableau.getWidth();
                 final int varStart = tableau.getNumObjectiveFunctions();
                 final int varEnd = tableau.getWidth() - 1;
                 for (Integer row : minRatioPositions) {
                     for (int i = varStart; i < varEnd && !row.equals(minRow); i++) {
                         final Integer basicRow = tableau.getBasicRow(i);
                         if (basicRow != null && basicRow.equals(row) && i < minIndex) {
                             minIndex = i;
                             minRow = row;
                         }
                     }
                 }
                 return minRow;
             }
         }
         return minRatioPositions.get(0);
     }
 
     /**
      * Runs one iteration of the Simplex method on the given model.
      *
      * @param tableau Simple tableau for the problem.
      * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
      * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
      */
     protected void doIteration(final SimplexTableau tableau)
         throws TooManyIterationsException,
                UnboundedSolutionException {
 
         incrementIterationCount();
 
         Integer pivotCol = getPivotColumn(tableau);
         Integer pivotRow = getPivotRow(tableau, pivotCol);
         if (pivotRow == null) {
             throw new UnboundedSolutionException();
         }
 
         // set the pivot element to 1
         double pivotVal = tableau.getEntry(pivotRow, pivotCol);
         tableau.divideRow(pivotRow, pivotVal);
 
         // set the rest of the pivot column to 0
         for (int i = 0; i < tableau.getHeight(); i++) {
             if (i != pivotRow) {
                 final double multiplier = tableau.getEntry(i, pivotCol);
                 tableau.subtractRow(i, pivotRow, multiplier);
             }
         }
     }
 
     /**
      * Solves Phase 1 of the Simplex method.
      *
      * @param tableau Simple tableau for the problem.
      * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
      * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
      * @throws NoFeasibleSolutionException if there is no feasible solution?
      */
     protected void solvePhase1(final SimplexTableau tableau)
         throws TooManyIterationsException,
                UnboundedSolutionException,
                NoFeasibleSolutionException {
 
         // make sure we're in Phase 1
         if (tableau.getNumArtificialVariables() == 0) {
             return;
         }
 
         while (!tableau.isOptimal()) {
             doIteration(tableau);
         }
 
         // if W is not zero then we have no feasible solution
         if (!Precision.equals(tableau.getEntry(0, tableau.getRhsOffset()), 0d, epsilon)) {
             throw new NoFeasibleSolutionException();
         }
     }
 
     /** {@inheritDoc} */
     @Override
     public PointValuePair doOptimize()
         throws TooManyIterationsException,
                UnboundedSolutionException,
                NoFeasibleSolutionException {
         final SimplexTableau tableau =
             new SimplexTableau(getFunction(),
                                getConstraints(),
                                getGoalType(),
                                isRestrictedToNonNegative(),
                                epsilon,
                                maxUlps,
                                cutOff);
 
         solvePhase1(tableau);
         tableau.dropPhase1Objective();
 
         while (!tableau.isOptimal()) {
             doIteration(tableau);
         }
         return tableau.getSolution();
     }
 }