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1   /*
2    * Licensed to the Apache Software Foundation (ASF) under one or more
3    * contributor license agreements.  See the NOTICE file distributed with
4    * this work for additional information regarding copyright ownership.
5    * The ASF licenses this file to You under the Apache License, Version 2.0
6    * (the "License"); you may not use this file except in compliance with
7    * the License.  You may obtain a copy of the License at
8    *
9    *      http://www.apache.org/licenses/LICENSE-2.0
10   *
11   * Unless required by applicable law or agreed to in writing, software
12   * distributed under the License is distributed on an "AS IS" BASIS,
13   * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
14   * See the License for the specific language governing permissions and
15   * limitations under the License.
16   */
17  package org.apache.commons.math3.optim.linear;
18  
19  import java.util.ArrayList;
20  import java.util.List;
21  import org.apache.commons.math3.exception.TooManyIterationsException;
22  import org.apache.commons.math3.optim.PointValuePair;
23  import org.apache.commons.math3.util.Precision;
24  
25  /**
26   * Solves a linear problem using the "Two-Phase Simplex" method.
27   * <p>
28   * <b>Note:</b> Depending on the problem definition, the default convergence criteria
29   * may be too strict, resulting in {@link NoFeasibleSolutionException} or
30   * {@link TooManyIterationsException}. In such a case it is advised to adjust these
31   * criteria with more appropriate values, e.g. relaxing the epsilon value.
32   * <p>
33   * Default convergence criteria:
34   * <ul>
35   *   <li>Algorithm convergence: 1e-6</li>
36   *   <li>Floating-point comparisons: 10 ulp</li>
37   *   <li>Cut-Off value: 1e-12</li>
38   * </ul>
39   * <p>
40   * The cut-off value has been introduced to zero out very small numbers in the Simplex tableau,
41   * as these may lead to numerical instabilities due to the nature of the Simplex algorithm
42   * (the pivot element is used as a denominator). If the problem definition is very tight, the
43   * default cut-off value may be too small, thus it is advised to increase it to a larger value,
44   * in accordance with the chosen epsilon.
45   * <p>
46   * It may also be counter-productive to provide a too large value for {@link
47   * org.apache.commons.math3.optim.MaxIter MaxIter} as parameter in the call of {@link
48   * #optimize(org.apache.commons.math3.optim.OptimizationData...) optimize(OptimizationData...)},
49   * as the {@link SimplexSolver} will use different strategies depending on the current iteration
50   * count. After half of the allowed max iterations has already been reached, the strategy to select
51   * pivot rows will change in order to break possible cycles due to degenerate problems.
52   *
53   * @version $Id: SimplexSolver.java 1462503 2013-03-29 15:48:27Z luc $
54   * @since 2.0
55   */
56  public class SimplexSolver extends LinearOptimizer {
57      /** Default amount of error to accept in floating point comparisons (as ulps). */
58      static final int DEFAULT_ULPS = 10;
59  
60      /** Default cut-off value. */
61      static final double DEFAULT_CUT_OFF = 1e-12;
62  
63      /** Default amount of error to accept for algorithm convergence. */
64      private static final double DEFAULT_EPSILON = 1.0e-6;
65  
66      /** Amount of error to accept for algorithm convergence. */
67      private final double epsilon;
68  
69      /** Amount of error to accept in floating point comparisons (as ulps). */
70      private final int maxUlps;
71  
72      /**
73       * Cut-off value for entries in the tableau: values smaller than the cut-off
74       * are treated as zero to improve numerical stability.
75       */
76      private final double cutOff;
77  
78      /**
79       * Builds a simplex solver with default settings.
80       */
81      public SimplexSolver() {
82          this(DEFAULT_EPSILON, DEFAULT_ULPS, DEFAULT_CUT_OFF);
83      }
84  
85      /**
86       * Builds a simplex solver with a specified accepted amount of error.
87       *
88       * @param epsilon Amount of error to accept for algorithm convergence.
89       */
90      public SimplexSolver(final double epsilon) {
91          this(epsilon, DEFAULT_ULPS, DEFAULT_CUT_OFF);
92      }
93  
94      /**
95       * Builds a simplex solver with a specified accepted amount of error.
96       *
97       * @param epsilon Amount of error to accept for algorithm convergence.
98       * @param maxUlps Amount of error to accept in floating point comparisons.
99       */
100     public SimplexSolver(final double epsilon, final int maxUlps) {
101         this(epsilon, maxUlps, DEFAULT_CUT_OFF);
102     }
103 
104     /**
105      * Builds a simplex solver with a specified accepted amount of error.
106      *
107      * @param epsilon Amount of error to accept for algorithm convergence.
108      * @param maxUlps Amount of error to accept in floating point comparisons.
109      * @param cutOff Values smaller than the cutOff are treated as zero.
110      */
111     public SimplexSolver(final double epsilon, final int maxUlps, final double cutOff) {
112         this.epsilon = epsilon;
113         this.maxUlps = maxUlps;
114         this.cutOff = cutOff;
115     }
116 
117     /**
118      * Returns the column with the most negative coefficient in the objective function row.
119      *
120      * @param tableau Simple tableau for the problem.
121      * @return the column with the most negative coefficient.
122      */
123     private Integer getPivotColumn(SimplexTableau tableau) {
124         double minValue = 0;
125         Integer minPos = null;
126         for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getWidth() - 1; i++) {
127             final double entry = tableau.getEntry(0, i);
128             // check if the entry is strictly smaller than the current minimum
129             // do not use a ulp/epsilon check
130             if (entry < minValue) {
131                 minValue = entry;
132                 minPos = i;
133             }
134         }
135         return minPos;
136     }
137 
138     /**
139      * Returns the row with the minimum ratio as given by the minimum ratio test (MRT).
140      *
141      * @param tableau Simple tableau for the problem.
142      * @param col Column to test the ratio of (see {@link #getPivotColumn(SimplexTableau)}).
143      * @return the row with the minimum ratio.
144      */
145     private Integer getPivotRow(SimplexTableau tableau, final int col) {
146         // create a list of all the rows that tie for the lowest score in the minimum ratio test
147         List<Integer> minRatioPositions = new ArrayList<Integer>();
148         double minRatio = Double.MAX_VALUE;
149         for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) {
150             final double rhs = tableau.getEntry(i, tableau.getWidth() - 1);
151             final double entry = tableau.getEntry(i, col);
152 
153             if (Precision.compareTo(entry, 0d, maxUlps) > 0) {
154                 final double ratio = rhs / entry;
155                 // check if the entry is strictly equal to the current min ratio
156                 // do not use a ulp/epsilon check
157                 final int cmp = Double.compare(ratio, minRatio);
158                 if (cmp == 0) {
159                     minRatioPositions.add(i);
160                 } else if (cmp < 0) {
161                     minRatio = ratio;
162                     minRatioPositions = new ArrayList<Integer>();
163                     minRatioPositions.add(i);
164                 }
165             }
166         }
167 
168         if (minRatioPositions.size() == 0) {
169             return null;
170         } else if (minRatioPositions.size() > 1) {
171             // there's a degeneracy as indicated by a tie in the minimum ratio test
172 
173             // 1. check if there's an artificial variable that can be forced out of the basis
174             if (tableau.getNumArtificialVariables() > 0) {
175                 for (Integer row : minRatioPositions) {
176                     for (int i = 0; i < tableau.getNumArtificialVariables(); i++) {
177                         int column = i + tableau.getArtificialVariableOffset();
178                         final double entry = tableau.getEntry(row, column);
179                         if (Precision.equals(entry, 1d, maxUlps) && row.equals(tableau.getBasicRow(column))) {
180                             return row;
181                         }
182                     }
183                 }
184             }
185 
186             // 2. apply Bland's rule to prevent cycling:
187             //    take the row for which the corresponding basic variable has the smallest index
188             //
189             // see http://www.stanford.edu/class/msande310/blandrule.pdf
190             // see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper)
191             //
192             // Additional heuristic: if we did not get a solution after half of maxIterations
193             //                       revert to the simple case of just returning the top-most row
194             // This heuristic is based on empirical data gathered while investigating MATH-828.
195             if (getEvaluations() < getMaxEvaluations() / 2) {
196                 Integer minRow = null;
197                 int minIndex = tableau.getWidth();
198                 final int varStart = tableau.getNumObjectiveFunctions();
199                 final int varEnd = tableau.getWidth() - 1;
200                 for (Integer row : minRatioPositions) {
201                     for (int i = varStart; i < varEnd && !row.equals(minRow); i++) {
202                         final Integer basicRow = tableau.getBasicRow(i);
203                         if (basicRow != null && basicRow.equals(row) && i < minIndex) {
204                             minIndex = i;
205                             minRow = row;
206                         }
207                     }
208                 }
209                 return minRow;
210             }
211         }
212         return minRatioPositions.get(0);
213     }
214 
215     /**
216      * Runs one iteration of the Simplex method on the given model.
217      *
218      * @param tableau Simple tableau for the problem.
219      * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
220      * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
221      */
222     protected void doIteration(final SimplexTableau tableau)
223         throws TooManyIterationsException,
224                UnboundedSolutionException {
225 
226         incrementIterationCount();
227 
228         Integer pivotCol = getPivotColumn(tableau);
229         Integer pivotRow = getPivotRow(tableau, pivotCol);
230         if (pivotRow == null) {
231             throw new UnboundedSolutionException();
232         }
233 
234         // set the pivot element to 1
235         double pivotVal = tableau.getEntry(pivotRow, pivotCol);
236         tableau.divideRow(pivotRow, pivotVal);
237 
238         // set the rest of the pivot column to 0
239         for (int i = 0; i < tableau.getHeight(); i++) {
240             if (i != pivotRow) {
241                 final double multiplier = tableau.getEntry(i, pivotCol);
242                 tableau.subtractRow(i, pivotRow, multiplier);
243             }
244         }
245     }
246 
247     /**
248      * Solves Phase 1 of the Simplex method.
249      *
250      * @param tableau Simple tableau for the problem.
251      * @throws TooManyIterationsException if the allowed number of iterations has been exhausted.
252      * @throws UnboundedSolutionException if the model is found not to have a bounded solution.
253      * @throws NoFeasibleSolutionException if there is no feasible solution?
254      */
255     protected void solvePhase1(final SimplexTableau tableau)
256         throws TooManyIterationsException,
257                UnboundedSolutionException,
258                NoFeasibleSolutionException {
259 
260         // make sure we're in Phase 1
261         if (tableau.getNumArtificialVariables() == 0) {
262             return;
263         }
264 
265         while (!tableau.isOptimal()) {
266             doIteration(tableau);
267         }
268 
269         // if W is not zero then we have no feasible solution
270         if (!Precision.equals(tableau.getEntry(0, tableau.getRhsOffset()), 0d, epsilon)) {
271             throw new NoFeasibleSolutionException();
272         }
273     }
274 
275     /** {@inheritDoc} */
276     @Override
277     public PointValuePair doOptimize()
278         throws TooManyIterationsException,
279                UnboundedSolutionException,
280                NoFeasibleSolutionException {
281         final SimplexTableau tableau =
282             new SimplexTableau(getFunction(),
283                                getConstraints(),
284                                getGoalType(),
285                                isRestrictedToNonNegative(),
286                                epsilon,
287                                maxUlps,
288                                cutOff);
289 
290         solvePhase1(tableau);
291         tableau.dropPhase1Objective();
292 
293         while (!tableau.isOptimal()) {
294             doIteration(tableau);
295         }
296         return tableau.getSolution();
297     }
298 }