001 /*
002 * Licensed to the Apache Software Foundation (ASF) under one or more
003 * contributor license agreements. See the NOTICE file distributed with
004 * this work for additional information regarding copyright ownership.
005 * The ASF licenses this file to You under the Apache License, Version 2.0
006 * (the "License"); you may not use this file except in compliance with
007 * the License. You may obtain a copy of the License at
008 *
009 * http://www.apache.org/licenses/LICENSE-2.0
010 *
011 * Unless required by applicable law or agreed to in writing, software
012 * distributed under the License is distributed on an "AS IS" BASIS,
013 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
014 * See the License for the specific language governing permissions and
015 * limitations under the License.
016 */
017 package org.apache.commons.math3.optim.linear;
018
019 import java.util.ArrayList;
020 import java.util.List;
021 import org.apache.commons.math3.exception.TooManyIterationsException;
022 import org.apache.commons.math3.optim.PointValuePair;
023 import org.apache.commons.math3.util.Precision;
024
025 /**
026 * Solves a linear problem using the "Two-Phase Simplex" method.
027 *
028 * @version $Id: SimplexSolver.java 1416643 2012-12-03 19:37:14Z tn $
029 * @since 2.0
030 */
031 public class SimplexSolver extends LinearOptimizer {
032 /** Default amount of error to accept for algorithm convergence. */
033 private static final double DEFAULT_EPSILON = 1.0e-6;
034
035 /** Default amount of error to accept in floating point comparisons (as ulps). */
036 private static final int DEFAULT_ULPS = 10;
037
038 /** Amount of error to accept for algorithm convergence. */
039 private final double epsilon;
040
041 /** Amount of error to accept in floating point comparisons (as ulps). */
042 private final int maxUlps;
043
044 /**
045 * Builds a simplex solver with default settings.
046 */
047 public SimplexSolver() {
048 this(DEFAULT_EPSILON, DEFAULT_ULPS);
049 }
050
051 /**
052 * Builds a simplex solver with a specified accepted amount of error.
053 *
054 * @param epsilon Amount of error to accept for algorithm convergence.
055 * @param maxUlps Amount of error to accept in floating point comparisons.
056 */
057 public SimplexSolver(final double epsilon,
058 final int maxUlps) {
059 this.epsilon = epsilon;
060 this.maxUlps = maxUlps;
061 }
062
063 /**
064 * Returns the column with the most negative coefficient in the objective function row.
065 *
066 * @param tableau Simple tableau for the problem.
067 * @return the column with the most negative coefficient.
068 */
069 private Integer getPivotColumn(SimplexTableau tableau) {
070 double minValue = 0;
071 Integer minPos = null;
072 for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getWidth() - 1; i++) {
073 final double entry = tableau.getEntry(0, i);
074 // check if the entry is strictly smaller than the current minimum
075 // do not use a ulp/epsilon check
076 if (entry < minValue) {
077 minValue = entry;
078 minPos = i;
079 }
080 }
081 return minPos;
082 }
083
084 /**
085 * Returns the row with the minimum ratio as given by the minimum ratio test (MRT).
086 *
087 * @param tableau Simple tableau for the problem.
088 * @param col Column to test the ratio of (see {@link #getPivotColumn(SimplexTableau)}).
089 * @return the row with the minimum ratio.
090 */
091 private Integer getPivotRow(SimplexTableau tableau, final int col) {
092 // create a list of all the rows that tie for the lowest score in the minimum ratio test
093 List<Integer> minRatioPositions = new ArrayList<Integer>();
094 double minRatio = Double.MAX_VALUE;
095 for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) {
096 final double rhs = tableau.getEntry(i, tableau.getWidth() - 1);
097 final double entry = tableau.getEntry(i, col);
098
099 if (Precision.compareTo(entry, 0d, maxUlps) > 0) {
100 final double ratio = rhs / entry;
101 // check if the entry is strictly equal to the current min ratio
102 // do not use a ulp/epsilon check
103 final int cmp = Double.compare(ratio, minRatio);
104 if (cmp == 0) {
105 minRatioPositions.add(i);
106 } else if (cmp < 0) {
107 minRatio = ratio;
108 minRatioPositions = new ArrayList<Integer>();
109 minRatioPositions.add(i);
110 }
111 }
112 }
113
114 if (minRatioPositions.size() == 0) {
115 return null;
116 } else if (minRatioPositions.size() > 1) {
117 // there's a degeneracy as indicated by a tie in the minimum ratio test
118
119 // 1. check if there's an artificial variable that can be forced out of the basis
120 if (tableau.getNumArtificialVariables() > 0) {
121 for (Integer row : minRatioPositions) {
122 for (int i = 0; i < tableau.getNumArtificialVariables(); i++) {
123 int column = i + tableau.getArtificialVariableOffset();
124 final double entry = tableau.getEntry(row, column);
125 if (Precision.equals(entry, 1d, maxUlps) && row.equals(tableau.getBasicRow(column))) {
126 return row;
127 }
128 }
129 }
130 }
131
132 // 2. apply Bland's rule to prevent cycling:
133 // take the row for which the corresponding basic variable has the smallest index
134 //
135 // see http://www.stanford.edu/class/msande310/blandrule.pdf
136 // see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper)
137 //
138 // Additional heuristic: if we did not get a solution after half of maxIterations
139 // revert to the simple case of just returning the top-most row
140 // This heuristic is based on empirical data gathered while investigating MATH-828.
141 if (getEvaluations() < getMaxEvaluations() / 2) {
142 Integer minRow = null;
143 int minIndex = tableau.getWidth();
144 final int varStart = tableau.getNumObjectiveFunctions();
145 final int varEnd = tableau.getWidth() - 1;
146 for (Integer row : minRatioPositions) {
147 for (int i = varStart; i < varEnd && !row.equals(minRow); i++) {
148 final Integer basicRow = tableau.getBasicRow(i);
149 if (basicRow != null && basicRow.equals(row)) {
150 if (i < minIndex) {
151 minIndex = i;
152 minRow = row;
153 }
154 }
155 }
156 }
157 return minRow;
158 }
159 }
160 return minRatioPositions.get(0);
161 }
162
163 /**
164 * Runs one iteration of the Simplex method on the given model.
165 *
166 * @param tableau Simple tableau for the problem.
167 * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
168 * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
169 */
170 protected void doIteration(final SimplexTableau tableau)
171 throws TooManyIterationsException,
172 UnboundedSolutionException {
173
174 incrementIterationCount();
175
176 Integer pivotCol = getPivotColumn(tableau);
177 Integer pivotRow = getPivotRow(tableau, pivotCol);
178 if (pivotRow == null) {
179 throw new UnboundedSolutionException();
180 }
181
182 // set the pivot element to 1
183 double pivotVal = tableau.getEntry(pivotRow, pivotCol);
184 tableau.divideRow(pivotRow, pivotVal);
185
186 // set the rest of the pivot column to 0
187 for (int i = 0; i < tableau.getHeight(); i++) {
188 if (i != pivotRow) {
189 final double multiplier = tableau.getEntry(i, pivotCol);
190 tableau.subtractRow(i, pivotRow, multiplier);
191 }
192 }
193 }
194
195 /**
196 * Solves Phase 1 of the Simplex method.
197 *
198 * @param tableau Simple tableau for the problem.
199 * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
200 * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
201 * @throws NoFeasibleSolutionException if there is no feasible solution?
202 */
203 protected void solvePhase1(final SimplexTableau tableau)
204 throws TooManyIterationsException,
205 UnboundedSolutionException,
206 NoFeasibleSolutionException {
207
208 // make sure we're in Phase 1
209 if (tableau.getNumArtificialVariables() == 0) {
210 return;
211 }
212
213 while (!tableau.isOptimal()) {
214 doIteration(tableau);
215 }
216
217 // if W is not zero then we have no feasible solution
218 if (!Precision.equals(tableau.getEntry(0, tableau.getRhsOffset()), 0d, epsilon)) {
219 throw new NoFeasibleSolutionException();
220 }
221 }
222
223 /** {@inheritDoc} */
224 @Override
225 public PointValuePair doOptimize()
226 throws TooManyIterationsException,
227 UnboundedSolutionException,
228 NoFeasibleSolutionException {
229 final SimplexTableau tableau =
230 new SimplexTableau(getFunction(),
231 getConstraints(),
232 getGoalType(),
233 isRestrictedToNonNegative(),
234 epsilon,
235 maxUlps);
236
237 solvePhase1(tableau);
238 tableau.dropPhase1Objective();
239
240 while (!tableau.isOptimal()) {
241 doIteration(tableau);
242 }
243 return tableau.getSolution();
244 }
245 }