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Diffstat (limited to 'src/main/java/org/apache/commons/math3/optim/linear/SimplexSolver.java')
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diff --git a/src/main/java/org/apache/commons/math3/optim/linear/SimplexSolver.java b/src/main/java/org/apache/commons/math3/optim/linear/SimplexSolver.java new file mode 100644 index 0000000..e95b657 --- /dev/null +++ b/src/main/java/org/apache/commons/math3/optim/linear/SimplexSolver.java @@ -0,0 +1,407 @@ +/* + * Licensed to the Apache Software Foundation (ASF) under one or more + * contributor license agreements. See the NOTICE file distributed with + * this work for additional information regarding copyright ownership. + * 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 + * the License. You may obtain a copy of the License at + * + * 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, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ +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; + } +} |