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    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
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    *
    *      http://www.apache.org/licenses/LICENSE-2.0
   *
   * Unless required by applicable law or agreed to in writing, software
   * distributed under the License is distributed on an "AS IS" BASIS,
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   * See the License for the specific language governing permissions and
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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.OptimizationData;
  import org.apache.commons.math3.optim.PointValuePair;
  import org.apache.commons.math3.util.FastMath;
  import org.apache.commons.math3.util.Precision;
  
  /**
   * Solves a linear problem using the "Two-Phase Simplex" method.
   * <p>
   * The {@link SimplexSolver} supports the following {@link OptimizationData} data provided
   * as arguments to {@link #optimize(OptimizationData...)}:
   * <ul>
   *   <li>objective function: {@link LinearObjectiveFunction} - mandatory</li>
   *   <li>linear constraints {@link LinearConstraintSet} - mandatory</li>
   *   <li>type of optimization: {@link org.apache.commons.math3.optim.nonlinear.scalar.GoalType GoalType}
   *    - optional, default: {@link org.apache.commons.math3.optim.nonlinear.scalar.GoalType#MINIMIZE MINIMIZE}</li>
   *   <li>whether to allow negative values as solution: {@link NonNegativeConstraint} - optional, default: true</li>
   *   <li>pivot selection rule: {@link PivotSelectionRule} - optional, default {@link PivotSelectionRule#DANTZIG}</li>
   *   <li>callback for the best solution: {@link SolutionCallback} - optional</li>
   *   <li>maximum number of iterations: {@link org.apache.commons.math3.optim.MaxIter} - optional, default: {@link Integer#MAX_VALUE}</li>
   * </ul>
   * <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-10</li>
    * </ul>
   * <p>
   * The cut-off value has been introduced to handle the case of very small pivot elements
   * in the Simplex tableau, as these may lead to numerical instabilities and degeneracy.
   * Potential pivot elements smaller than this value will be treated as if they were zero
   * and are thus not considered by the pivot selection mechanism. The default value is safe
   * for many problems, but may need to be adjusted in case of very small coefficients
   * used in either the {@link LinearConstraint} or {@link LinearObjectiveFunction}.
   *
   * @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-10;
  
      /** 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;
  
      /** The pivot selection method to use. */
      private PivotSelectionRule pivotSelection;
  
      /**
       * The solution callback to access the best solution found so far in case
       * the optimizer fails to find an optimal solution within the iteration limits.
       */
      private SolutionCallback solutionCallback;
  
      /**
       * 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;
         this.pivotSelection = PivotSelectionRule.DANTZIG;
     }
 
     /**
      * {@inheritDoc}
      *
      * @param optData Optimization data. In addition to those documented in
      * {@link LinearOptimizer#optimize(OptimizationData...)
      * LinearOptimizer}, this method will register the following data:
      * <ul>
      *  <li>{@link SolutionCallback}</li>
      *  <li>{@link PivotSelectionRule}</li>
      * </ul>
      *
      * @return {@inheritDoc}
      * @throws TooManyIterationsException if the maximal number of iterations is exceeded.
      */
     @Override
     public PointValuePair optimize(OptimizationData... optData)
         throws TooManyIterationsException {
         // Set up base class and perform computation.
         return super.optimize(optData);
     }
 
     /**
      * {@inheritDoc}
      *
      * @param optData Optimization data.
      * In addition to those documented in
      * {@link LinearOptimizer#parseOptimizationData(OptimizationData[])
      * LinearOptimizer}, this method will register the following data:
      * <ul>
      *  <li>{@link SolutionCallback}</li>
      *  <li>{@link PivotSelectionRule}</li>
      * </ul>
      */
     @Override
     protected void parseOptimizationData(OptimizationData... optData) {
         // Allow base class to register its own data.
         super.parseOptimizationData(optData);
 
         // reset the callback before parsing
         solutionCallback = null;
 
         for (OptimizationData data : optData) {
             if (data instanceof SolutionCallback) {
                 solutionCallback = (SolutionCallback) data;
                 continue;
             }
             if (data instanceof PivotSelectionRule) {
                 pivotSelection = (PivotSelectionRule) data;
                 continue;
             }
         }
     }
 
     /**
      * 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;
 
                 // Bland's rule: chose the entering column with the lowest index
                 if (pivotSelection == PivotSelectionRule.BLAND && isValidPivotColumn(tableau, i)) {
                     break;
                 }
             }
         }
         return minPos;
     }
 
     /**
      * Checks whether the given column is valid pivot column, i.e. will result
      * in a valid pivot row.
      * <p>
      * When applying Bland's rule to select the pivot column, it may happen that
      * there is no corresponding pivot row. This method will check if the selected
      * pivot column will return a valid pivot row.
      *
      * @param tableau simplex tableau for the problem
      * @param col the column to test
      * @return {@code true} if the pivot column is valid, {@code false} otherwise
      */
     private boolean isValidPivotColumn(SimplexTableau tableau, int col) {
         for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) {
             final double entry = tableau.getEntry(i, col);
 
             // do the same check as in getPivotRow
             if (Precision.compareTo(entry, 0d, cutOff) > 0) {
                 return true;
             }
         }
         return false;
     }
 
     /**
      * Returns the row with the minimum ratio as given by the minimum ratio test (MRT).
      *
      * @param tableau Simplex 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);
 
             // only consider pivot elements larger than the cutOff threshold
             // selecting others may lead to degeneracy or numerical instabilities
             if (Precision.compareTo(entry, 0d, cutOff) > 0) {
                 final double ratio = FastMath.abs(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.clear();
                     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)
 
             Integer minRow = null;
             int minIndex = tableau.getWidth();
             for (Integer row : minRatioPositions) {
                 final int basicVar = tableau.getBasicVariable(row);
                 if (basicVar < minIndex) {
                     minIndex = basicVar;
                     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();
         }
 
         tableau.performRowOperations(pivotCol, pivotRow);
     }
 
     /**
      * 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 {
 
         // reset the tableau to indicate a non-feasible solution in case
         // we do not pass phase 1 successfully
         if (solutionCallback != null) {
             solutionCallback.setTableau(null);
         }
 
         final SimplexTableau tableau =
             new SimplexTableau(getFunction(),
                                getConstraints(),
                                getGoalType(),
                                isRestrictedToNonNegative(),
                                epsilon,
                                maxUlps);
 
         solvePhase1(tableau);
         tableau.dropPhase1Objective();
 
         // after phase 1, we are sure to have a feasible solution
         if (solutionCallback != null) {
             solutionCallback.setTableau(tableau);
         }
 
         while (!tableau.isOptimal()) {
             doIteration(tableau);
         }
 
         // check that the solution respects the nonNegative restriction in case
         // the epsilon/cutOff values are too large for the actual linear problem
         // (e.g. with very small constraint coefficients), the solver might actually
         // find a non-valid solution (with negative coefficients).
         final PointValuePair solution = tableau.getSolution();
         if (isRestrictedToNonNegative()) {
             final double[] coeff = solution.getPoint();
             for (int i = 0; i < coeff.length; i++) {
                 if (Precision.compareTo(coeff[i], 0, epsilon) < 0) {
                     throw new NoFeasibleSolutionException();
                 }
             }
         }
         return solution;
     }
 }