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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* 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.PointValuePair;
import org.apache.commons.math3.util.Precision;
/**
* Solves a linear problem using the "Two-Phase Simplex" method.
*
* @version $Id: SimplexSolver.java 1416643 2012-12-03 19:37:14Z tn $
* @since 2.0
*/
public class SimplexSolver extends LinearOptimizer {
/** Default amount of error to accept for algorithm convergence. */
private static final double DEFAULT_EPSILON = 1.0e-6;
/** Default amount of error to accept in floating point comparisons (as ulps). */
private static final int DEFAULT_ULPS = 10;
/** 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;
/**
* Builds a simplex solver with default settings.
*/
public SimplexSolver() {
this(DEFAULT_EPSILON, DEFAULT_ULPS);
}
/**
* 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 = epsilon;
this.maxUlps = maxUlps;
}
/**
* 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 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);
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();
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)) {
if (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);
solvePhase1(tableau);
tableau.dropPhase1Objective();
while (!tableau.isOptimal()) {
doIteration(tableau);
}
return tableau.getSolution();
}
}
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