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* 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
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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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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;
}
}