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 */ 017package org.apache.commons.math3.optim.linear; 018 019import java.util.ArrayList; 020import java.util.List; 021 022import org.apache.commons.math3.exception.TooManyIterationsException; 023import org.apache.commons.math3.optim.OptimizationData; 024import org.apache.commons.math3.optim.PointValuePair; 025import org.apache.commons.math3.util.FastMath; 026import org.apache.commons.math3.util.Precision; 027 028/** 029 * Solves a linear problem using the "Two-Phase Simplex" method. 030 * <p> 031 * The {@link SimplexSolver} supports the following {@link OptimizationData} data provided 032 * as arguments to {@link #optimize(OptimizationData...)}: 033 * <ul> 034 * <li>objective function: {@link LinearObjectiveFunction} - mandatory</li> 035 * <li>linear constraints {@link LinearConstraintSet} - mandatory</li> 036 * <li>type of optimization: {@link org.apache.commons.math3.optim.nonlinear.scalar.GoalType GoalType} 037 * - optional, default: {@link org.apache.commons.math3.optim.nonlinear.scalar.GoalType#MINIMIZE MINIMIZE}</li> 038 * <li>whether to allow negative values as solution: {@link NonNegativeConstraint} - optional, default: true</li> 039 * <li>pivot selection rule: {@link PivotSelectionRule} - optional, default {@link PivotSelectionRule#DANTZIG}</li> 040 * <li>callback for the best solution: {@link SolutionCallback} - optional</li> 041 * <li>maximum number of iterations: {@link org.apache.commons.math3.optim.MaxIter} - optional, default: {@link Integer#MAX_VALUE}</li> 042 * </ul> 043 * <p> 044 * <b>Note:</b> Depending on the problem definition, the default convergence criteria 045 * may be too strict, resulting in {@link NoFeasibleSolutionException} or 046 * {@link TooManyIterationsException}. In such a case it is advised to adjust these 047 * criteria with more appropriate values, e.g. relaxing the epsilon value. 048 * <p> 049 * Default convergence criteria: 050 * <ul> 051 * <li>Algorithm convergence: 1e-6</li> 052 * <li>Floating-point comparisons: 10 ulp</li> 053 * <li>Cut-Off value: 1e-10</li> 054 * </ul> 055 * <p> 056 * The cut-off value has been introduced to handle the case of very small pivot elements 057 * in the Simplex tableau, as these may lead to numerical instabilities and degeneracy. 058 * Potential pivot elements smaller than this value will be treated as if they were zero 059 * and are thus not considered by the pivot selection mechanism. The default value is safe 060 * for many problems, but may need to be adjusted in case of very small coefficients 061 * used in either the {@link LinearConstraint} or {@link LinearObjectiveFunction}. 062 * 063 * @since 2.0 064 */ 065public class SimplexSolver extends LinearOptimizer { 066 /** Default amount of error to accept in floating point comparisons (as ulps). */ 067 static final int DEFAULT_ULPS = 10; 068 069 /** Default cut-off value. */ 070 static final double DEFAULT_CUT_OFF = 1e-10; 071 072 /** Default amount of error to accept for algorithm convergence. */ 073 private static final double DEFAULT_EPSILON = 1.0e-6; 074 075 /** Amount of error to accept for algorithm convergence. */ 076 private final double epsilon; 077 078 /** Amount of error to accept in floating point comparisons (as ulps). */ 079 private final int maxUlps; 080 081 /** 082 * Cut-off value for entries in the tableau: values smaller than the cut-off 083 * are treated as zero to improve numerical stability. 084 */ 085 private final double cutOff; 086 087 /** The pivot selection method to use. */ 088 private PivotSelectionRule pivotSelection; 089 090 /** 091 * The solution callback to access the best solution found so far in case 092 * the optimizer fails to find an optimal solution within the iteration limits. 093 */ 094 private SolutionCallback solutionCallback; 095 096 /** 097 * Builds a simplex solver with default settings. 098 */ 099 public SimplexSolver() { 100 this(DEFAULT_EPSILON, DEFAULT_ULPS, DEFAULT_CUT_OFF); 101 } 102 103 /** 104 * Builds a simplex solver with a specified accepted amount of error. 105 * 106 * @param epsilon Amount of error to accept for algorithm convergence. 107 */ 108 public SimplexSolver(final double epsilon) { 109 this(epsilon, DEFAULT_ULPS, DEFAULT_CUT_OFF); 110 } 111 112 /** 113 * Builds a simplex solver with a specified accepted amount of error. 114 * 115 * @param epsilon Amount of error to accept for algorithm convergence. 116 * @param maxUlps Amount of error to accept in floating point comparisons. 117 */ 118 public SimplexSolver(final double epsilon, final int maxUlps) { 119 this(epsilon, maxUlps, DEFAULT_CUT_OFF); 120 } 121 122 /** 123 * Builds a simplex solver with a specified accepted amount of error. 124 * 125 * @param epsilon Amount of error to accept for algorithm convergence. 126 * @param maxUlps Amount of error to accept in floating point comparisons. 127 * @param cutOff Values smaller than the cutOff are treated as zero. 128 */ 129 public SimplexSolver(final double epsilon, final int maxUlps, final double cutOff) { 130 this.epsilon = epsilon; 131 this.maxUlps = maxUlps; 132 this.cutOff = cutOff; 133 this.pivotSelection = PivotSelectionRule.DANTZIG; 134 } 135 136 /** 137 * {@inheritDoc} 138 * 139 * @param optData Optimization data. In addition to those documented in 140 * {@link LinearOptimizer#optimize(OptimizationData...) 141 * LinearOptimizer}, this method will register the following data: 142 * <ul> 143 * <li>{@link SolutionCallback}</li> 144 * <li>{@link PivotSelectionRule}</li> 145 * </ul> 146 * 147 * @return {@inheritDoc} 148 * @throws TooManyIterationsException if the maximal number of iterations is exceeded. 149 */ 150 @Override 151 public PointValuePair optimize(OptimizationData... optData) 152 throws TooManyIterationsException { 153 // Set up base class and perform computation. 154 return super.optimize(optData); 155 } 156 157 /** 158 * {@inheritDoc} 159 * 160 * @param optData Optimization data. 161 * In addition to those documented in 162 * {@link LinearOptimizer#parseOptimizationData(OptimizationData[]) 163 * LinearOptimizer}, this method will register the following data: 164 * <ul> 165 * <li>{@link SolutionCallback}</li> 166 * <li>{@link PivotSelectionRule}</li> 167 * </ul> 168 */ 169 @Override 170 protected void parseOptimizationData(OptimizationData... optData) { 171 // Allow base class to register its own data. 172 super.parseOptimizationData(optData); 173 174 // reset the callback before parsing 175 solutionCallback = null; 176 177 for (OptimizationData data : optData) { 178 if (data instanceof SolutionCallback) { 179 solutionCallback = (SolutionCallback) data; 180 continue; 181 } 182 if (data instanceof PivotSelectionRule) { 183 pivotSelection = (PivotSelectionRule) data; 184 continue; 185 } 186 } 187 } 188 189 /** 190 * Returns the column with the most negative coefficient in the objective function row. 191 * 192 * @param tableau Simple tableau for the problem. 193 * @return the column with the most negative coefficient. 194 */ 195 private Integer getPivotColumn(SimplexTableau tableau) { 196 double minValue = 0; 197 Integer minPos = null; 198 for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getWidth() - 1; i++) { 199 final double entry = tableau.getEntry(0, i); 200 // check if the entry is strictly smaller than the current minimum 201 // do not use a ulp/epsilon check 202 if (entry < minValue) { 203 minValue = entry; 204 minPos = i; 205 206 // Bland's rule: chose the entering column with the lowest index 207 if (pivotSelection == PivotSelectionRule.BLAND && isValidPivotColumn(tableau, i)) { 208 break; 209 } 210 } 211 } 212 return minPos; 213 } 214 215 /** 216 * Checks whether the given column is valid pivot column, i.e. will result 217 * in a valid pivot row. 218 * <p> 219 * When applying Bland's rule to select the pivot column, it may happen that 220 * there is no corresponding pivot row. This method will check if the selected 221 * pivot column will return a valid pivot row. 222 * 223 * @param tableau simplex tableau for the problem 224 * @param col the column to test 225 * @return {@code true} if the pivot column is valid, {@code false} otherwise 226 */ 227 private boolean isValidPivotColumn(SimplexTableau tableau, int col) { 228 for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) { 229 final double entry = tableau.getEntry(i, col); 230 231 // do the same check as in getPivotRow 232 if (Precision.compareTo(entry, 0d, cutOff) > 0) { 233 return true; 234 } 235 } 236 return false; 237 } 238 239 /** 240 * Returns the row with the minimum ratio as given by the minimum ratio test (MRT). 241 * 242 * @param tableau Simplex tableau for the problem. 243 * @param col Column to test the ratio of (see {@link #getPivotColumn(SimplexTableau)}). 244 * @return the row with the minimum ratio. 245 */ 246 private Integer getPivotRow(SimplexTableau tableau, final int col) { 247 // create a list of all the rows that tie for the lowest score in the minimum ratio test 248 List<Integer> minRatioPositions = new ArrayList<Integer>(); 249 double minRatio = Double.MAX_VALUE; 250 for (int i = tableau.getNumObjectiveFunctions(); i < tableau.getHeight(); i++) { 251 final double rhs = tableau.getEntry(i, tableau.getWidth() - 1); 252 final double entry = tableau.getEntry(i, col); 253 254 // only consider pivot elements larger than the cutOff threshold 255 // selecting others may lead to degeneracy or numerical instabilities 256 if (Precision.compareTo(entry, 0d, cutOff) > 0) { 257 final double ratio = FastMath.abs(rhs / entry); 258 // check if the entry is strictly equal to the current min ratio 259 // do not use a ulp/epsilon check 260 final int cmp = Double.compare(ratio, minRatio); 261 if (cmp == 0) { 262 minRatioPositions.add(i); 263 } else if (cmp < 0) { 264 minRatio = ratio; 265 minRatioPositions.clear(); 266 minRatioPositions.add(i); 267 } 268 } 269 } 270 271 if (minRatioPositions.size() == 0) { 272 return null; 273 } else if (minRatioPositions.size() > 1) { 274 // there's a degeneracy as indicated by a tie in the minimum ratio test 275 276 // 1. check if there's an artificial variable that can be forced out of the basis 277 if (tableau.getNumArtificialVariables() > 0) { 278 for (Integer row : minRatioPositions) { 279 for (int i = 0; i < tableau.getNumArtificialVariables(); i++) { 280 int column = i + tableau.getArtificialVariableOffset(); 281 final double entry = tableau.getEntry(row, column); 282 if (Precision.equals(entry, 1d, maxUlps) && row.equals(tableau.getBasicRow(column))) { 283 return row; 284 } 285 } 286 } 287 } 288 289 // 2. apply Bland's rule to prevent cycling: 290 // take the row for which the corresponding basic variable has the smallest index 291 // 292 // see http://www.stanford.edu/class/msande310/blandrule.pdf 293 // see http://en.wikipedia.org/wiki/Bland%27s_rule (not equivalent to the above paper) 294 295 Integer minRow = null; 296 int minIndex = tableau.getWidth(); 297 for (Integer row : minRatioPositions) { 298 final int basicVar = tableau.getBasicVariable(row); 299 if (basicVar < minIndex) { 300 minIndex = basicVar; 301 minRow = row; 302 } 303 } 304 return minRow; 305 } 306 return minRatioPositions.get(0); 307 } 308 309 /** 310 * Runs one iteration of the Simplex method on the given model. 311 * 312 * @param tableau Simple tableau for the problem. 313 * @throws TooManyIterationsException if the allowed number of iterations has been exhausted. 314 * @throws UnboundedSolutionException if the model is found not to have a bounded solution. 315 */ 316 protected void doIteration(final SimplexTableau tableau) 317 throws TooManyIterationsException, 318 UnboundedSolutionException { 319 320 incrementIterationCount(); 321 322 Integer pivotCol = getPivotColumn(tableau); 323 Integer pivotRow = getPivotRow(tableau, pivotCol); 324 if (pivotRow == null) { 325 throw new UnboundedSolutionException(); 326 } 327 328 tableau.performRowOperations(pivotCol, pivotRow); 329 } 330 331 /** 332 * Solves Phase 1 of the Simplex method. 333 * 334 * @param tableau Simple tableau for the problem. 335 * @throws TooManyIterationsException if the allowed number of iterations has been exhausted. 336 * @throws UnboundedSolutionException if the model is found not to have a bounded solution. 337 * @throws NoFeasibleSolutionException if there is no feasible solution? 338 */ 339 protected void solvePhase1(final SimplexTableau tableau) 340 throws TooManyIterationsException, 341 UnboundedSolutionException, 342 NoFeasibleSolutionException { 343 344 // make sure we're in Phase 1 345 if (tableau.getNumArtificialVariables() == 0) { 346 return; 347 } 348 349 while (!tableau.isOptimal()) { 350 doIteration(tableau); 351 } 352 353 // if W is not zero then we have no feasible solution 354 if (!Precision.equals(tableau.getEntry(0, tableau.getRhsOffset()), 0d, epsilon)) { 355 throw new NoFeasibleSolutionException(); 356 } 357 } 358 359 /** {@inheritDoc} */ 360 @Override 361 public PointValuePair doOptimize() 362 throws TooManyIterationsException, 363 UnboundedSolutionException, 364 NoFeasibleSolutionException { 365 366 // reset the tableau to indicate a non-feasible solution in case 367 // we do not pass phase 1 successfully 368 if (solutionCallback != null) { 369 solutionCallback.setTableau(null); 370 } 371 372 final SimplexTableau tableau = 373 new SimplexTableau(getFunction(), 374 getConstraints(), 375 getGoalType(), 376 isRestrictedToNonNegative(), 377 epsilon, 378 maxUlps); 379 380 solvePhase1(tableau); 381 tableau.dropPhase1Objective(); 382 383 // after phase 1, we are sure to have a feasible solution 384 if (solutionCallback != null) { 385 solutionCallback.setTableau(tableau); 386 } 387 388 while (!tableau.isOptimal()) { 389 doIteration(tableau); 390 } 391 392 // check that the solution respects the nonNegative restriction in case 393 // the epsilon/cutOff values are too large for the actual linear problem 394 // (e.g. with very small constraint coefficients), the solver might actually 395 // find a non-valid solution (with negative coefficients). 396 final PointValuePair solution = tableau.getSolution(); 397 if (isRestrictedToNonNegative()) { 398 final double[] coeff = solution.getPoint(); 399 for (int i = 0; i < coeff.length; i++) { 400 if (Precision.compareTo(coeff[i], 0, epsilon) < 0) { 401 throw new NoFeasibleSolutionException(); 402 } 403 } 404 } 405 return solution; 406 } 407}