SimplexSolver.java
- /*
- * 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
- *
- * https://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.
- */
- /*
- * This is not the original file distributed by the Apache Software Foundation
- * It has been modified by the Hipparchus project
- */
- package org.hipparchus.optim.linear;
- import java.util.ArrayList;
- import java.util.List;
- import org.hipparchus.exception.MathIllegalStateException;
- import org.hipparchus.optim.LocalizedOptimFormats;
- import org.hipparchus.optim.OptimizationData;
- import org.hipparchus.optim.PointValuePair;
- import org.hipparchus.util.FastMath;
- import org.hipparchus.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.hipparchus.optim.nonlinear.scalar.GoalType GoalType}
- * - optional, default: {@link org.hipparchus.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.hipparchus.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 MathIllegalStateException} or
- * {@link MathIllegalStateException}. 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}.
- *
- */
- 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 MathIllegalStateException if the maximal number of iterations is exceeded.
- * @throws org.hipparchus.exception.MathIllegalArgumentException if the dimension
- * of the constraints does not match the dimension of the objective function
- */
- @Override
- public PointValuePair optimize(OptimizationData... optData)
- throws MathIllegalStateException {
- // 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<>();
- 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.isEmpty()) {
- 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 MathIllegalStateException if the allowed number of iterations has been exhausted.
- * @throws MathIllegalStateException if the model is found not to have a bounded solution.
- */
- protected void doIteration(final SimplexTableau tableau)
- throws MathIllegalStateException {
- incrementIterationCount();
- Integer pivotCol = getPivotColumn(tableau);
- Integer pivotRow = getPivotRow(tableau, pivotCol);
- if (pivotRow == null) {
- throw new MathIllegalStateException(LocalizedOptimFormats.UNBOUNDED_SOLUTION);
- }
- tableau.performRowOperations(pivotCol, pivotRow);
- }
- /**
- * Solves Phase 1 of the Simplex method.
- *
- * @param tableau Simple tableau for the problem.
- * @throws MathIllegalStateException if the allowed number of iterations has been exhausted,
- * or if the model is found not to have a bounded solution, or if there is no feasible solution
- */
- protected void solvePhase1(final SimplexTableau tableau)
- throws MathIllegalStateException {
- // 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 MathIllegalStateException(LocalizedOptimFormats.NO_FEASIBLE_SOLUTION);
- }
- }
- /** {@inheritDoc} */
- @Override
- public PointValuePair doOptimize()
- throws MathIllegalStateException {
- // 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 MathIllegalStateException(LocalizedOptimFormats.NO_FEASIBLE_SOLUTION);
- }
- }
- }
- return solution;
- }
- }