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.nonlinear.scalar.noderiv;
018
019 import org.apache.commons.math3.util.FastMath;
020 import org.apache.commons.math3.util.MathArrays;
021 import org.apache.commons.math3.analysis.UnivariateFunction;
022 import org.apache.commons.math3.exception.NumberIsTooSmallException;
023 import org.apache.commons.math3.exception.NotStrictlyPositiveException;
024 import org.apache.commons.math3.optim.MaxEval;
025 import org.apache.commons.math3.optim.nonlinear.scalar.GoalType;
026 import org.apache.commons.math3.optim.PointValuePair;
027 import org.apache.commons.math3.optim.ConvergenceChecker;
028 import org.apache.commons.math3.optim.nonlinear.scalar.MultivariateOptimizer;
029 import org.apache.commons.math3.optim.univariate.BracketFinder;
030 import org.apache.commons.math3.optim.univariate.BrentOptimizer;
031 import org.apache.commons.math3.optim.univariate.UnivariatePointValuePair;
032 import org.apache.commons.math3.optim.univariate.SimpleUnivariateValueChecker;
033 import org.apache.commons.math3.optim.univariate.SearchInterval;
034 import org.apache.commons.math3.optim.univariate.UnivariateObjectiveFunction;
035
036 /**
037 * Powell algorithm.
038 * This code is translated and adapted from the Python version of this
039 * algorithm (as implemented in module {@code optimize.py} v0.5 of
040 * <em>SciPy</em>).
041 * <br/>
042 * The default stopping criterion is based on the differences of the
043 * function value between two successive iterations. It is however possible
044 * to define a custom convergence checker that might terminate the algorithm
045 * earlier.
046 * <br/>
047 * The internal line search optimizer is a {@link BrentOptimizer} with a
048 * convergence checker set to {@link SimpleUnivariateValueChecker}.
049 *
050 * @version $Id: PowellOptimizer.java 1413594 2012-11-26 13:16:39Z erans $
051 * @since 2.2
052 */
053 public class PowellOptimizer
054 extends MultivariateOptimizer {
055 /**
056 * Minimum relative tolerance.
057 */
058 private static final double MIN_RELATIVE_TOLERANCE = 2 * FastMath.ulp(1d);
059 /**
060 * Relative threshold.
061 */
062 private final double relativeThreshold;
063 /**
064 * Absolute threshold.
065 */
066 private final double absoluteThreshold;
067 /**
068 * Line search.
069 */
070 private final LineSearch line;
071
072 /**
073 * This constructor allows to specify a user-defined convergence checker,
074 * in addition to the parameters that control the default convergence
075 * checking procedure.
076 * <br/>
077 * The internal line search tolerances are set to the square-root of their
078 * corresponding value in the multivariate optimizer.
079 *
080 * @param rel Relative threshold.
081 * @param abs Absolute threshold.
082 * @param checker Convergence checker.
083 * @throws NotStrictlyPositiveException if {@code abs <= 0}.
084 * @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
085 */
086 public PowellOptimizer(double rel,
087 double abs,
088 ConvergenceChecker<PointValuePair> checker) {
089 this(rel, abs, FastMath.sqrt(rel), FastMath.sqrt(abs), checker);
090 }
091
092 /**
093 * This constructor allows to specify a user-defined convergence checker,
094 * in addition to the parameters that control the default convergence
095 * checking procedure and the line search tolerances.
096 *
097 * @param rel Relative threshold for this optimizer.
098 * @param abs Absolute threshold for this optimizer.
099 * @param lineRel Relative threshold for the internal line search optimizer.
100 * @param lineAbs Absolute threshold for the internal line search optimizer.
101 * @param checker Convergence checker.
102 * @throws NotStrictlyPositiveException if {@code abs <= 0}.
103 * @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
104 */
105 public PowellOptimizer(double rel,
106 double abs,
107 double lineRel,
108 double lineAbs,
109 ConvergenceChecker<PointValuePair> checker) {
110 super(checker);
111
112 if (rel < MIN_RELATIVE_TOLERANCE) {
113 throw new NumberIsTooSmallException(rel, MIN_RELATIVE_TOLERANCE, true);
114 }
115 if (abs <= 0) {
116 throw new NotStrictlyPositiveException(abs);
117 }
118 relativeThreshold = rel;
119 absoluteThreshold = abs;
120
121 // Create the line search optimizer.
122 line = new LineSearch(lineRel,
123 lineAbs);
124 }
125
126 /**
127 * The parameters control the default convergence checking procedure.
128 * <br/>
129 * The internal line search tolerances are set to the square-root of their
130 * corresponding value in the multivariate optimizer.
131 *
132 * @param rel Relative threshold.
133 * @param abs Absolute threshold.
134 * @throws NotStrictlyPositiveException if {@code abs <= 0}.
135 * @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
136 */
137 public PowellOptimizer(double rel,
138 double abs) {
139 this(rel, abs, null);
140 }
141
142 /**
143 * Builds an instance with the default convergence checking procedure.
144 *
145 * @param rel Relative threshold.
146 * @param abs Absolute threshold.
147 * @param lineRel Relative threshold for the internal line search optimizer.
148 * @param lineAbs Absolute threshold for the internal line search optimizer.
149 * @throws NotStrictlyPositiveException if {@code abs <= 0}.
150 * @throws NumberIsTooSmallException if {@code rel < 2 * Math.ulp(1d)}.
151 */
152 public PowellOptimizer(double rel,
153 double abs,
154 double lineRel,
155 double lineAbs) {
156 this(rel, abs, lineRel, lineAbs, null);
157 }
158
159 /** {@inheritDoc} */
160 @Override
161 protected PointValuePair doOptimize() {
162 final GoalType goal = getGoalType();
163 final double[] guess = getStartPoint();
164 final int n = guess.length;
165
166 final double[][] direc = new double[n][n];
167 for (int i = 0; i < n; i++) {
168 direc[i][i] = 1;
169 }
170
171 final ConvergenceChecker<PointValuePair> checker
172 = getConvergenceChecker();
173
174 double[] x = guess;
175 double fVal = computeObjectiveValue(x);
176 double[] x1 = x.clone();
177 int iter = 0;
178 while (true) {
179 ++iter;
180
181 double fX = fVal;
182 double fX2 = 0;
183 double delta = 0;
184 int bigInd = 0;
185 double alphaMin = 0;
186
187 for (int i = 0; i < n; i++) {
188 final double[] d = MathArrays.copyOf(direc[i]);
189
190 fX2 = fVal;
191
192 final UnivariatePointValuePair optimum = line.search(x, d);
193 fVal = optimum.getValue();
194 alphaMin = optimum.getPoint();
195 final double[][] result = newPointAndDirection(x, d, alphaMin);
196 x = result[0];
197
198 if ((fX2 - fVal) > delta) {
199 delta = fX2 - fVal;
200 bigInd = i;
201 }
202 }
203
204 // Default convergence check.
205 boolean stop = 2 * (fX - fVal) <=
206 (relativeThreshold * (FastMath.abs(fX) + FastMath.abs(fVal)) +
207 absoluteThreshold);
208
209 final PointValuePair previous = new PointValuePair(x1, fX);
210 final PointValuePair current = new PointValuePair(x, fVal);
211 if (!stop) { // User-defined stopping criteria.
212 if (checker != null) {
213 stop = checker.converged(iter, previous, current);
214 }
215 }
216 if (stop) {
217 if (goal == GoalType.MINIMIZE) {
218 return (fVal < fX) ? current : previous;
219 } else {
220 return (fVal > fX) ? current : previous;
221 }
222 }
223
224 final double[] d = new double[n];
225 final double[] x2 = new double[n];
226 for (int i = 0; i < n; i++) {
227 d[i] = x[i] - x1[i];
228 x2[i] = 2 * x[i] - x1[i];
229 }
230
231 x1 = x.clone();
232 fX2 = computeObjectiveValue(x2);
233
234 if (fX > fX2) {
235 double t = 2 * (fX + fX2 - 2 * fVal);
236 double temp = fX - fVal - delta;
237 t *= temp * temp;
238 temp = fX - fX2;
239 t -= delta * temp * temp;
240
241 if (t < 0.0) {
242 final UnivariatePointValuePair optimum = line.search(x, d);
243 fVal = optimum.getValue();
244 alphaMin = optimum.getPoint();
245 final double[][] result = newPointAndDirection(x, d, alphaMin);
246 x = result[0];
247
248 final int lastInd = n - 1;
249 direc[bigInd] = direc[lastInd];
250 direc[lastInd] = result[1];
251 }
252 }
253 }
254 }
255
256 /**
257 * Compute a new point (in the original space) and a new direction
258 * vector, resulting from the line search.
259 *
260 * @param p Point used in the line search.
261 * @param d Direction used in the line search.
262 * @param optimum Optimum found by the line search.
263 * @return a 2-element array containing the new point (at index 0) and
264 * the new direction (at index 1).
265 */
266 private double[][] newPointAndDirection(double[] p,
267 double[] d,
268 double optimum) {
269 final int n = p.length;
270 final double[] nP = new double[n];
271 final double[] nD = new double[n];
272 for (int i = 0; i < n; i++) {
273 nD[i] = d[i] * optimum;
274 nP[i] = p[i] + nD[i];
275 }
276
277 final double[][] result = new double[2][];
278 result[0] = nP;
279 result[1] = nD;
280
281 return result;
282 }
283
284 /**
285 * Class for finding the minimum of the objective function along a given
286 * direction.
287 */
288 private class LineSearch extends BrentOptimizer {
289 /**
290 * Value that will pass the precondition check for {@link BrentOptimizer}
291 * but will not pass the convergence check, so that the custom checker
292 * will always decide when to stop the line search.
293 */
294 private static final double REL_TOL_UNUSED = 1e-15;
295 /**
296 * Value that will pass the precondition check for {@link BrentOptimizer}
297 * but will not pass the convergence check, so that the custom checker
298 * will always decide when to stop the line search.
299 */
300 private static final double ABS_TOL_UNUSED = Double.MIN_VALUE;
301 /**
302 * Automatic bracketing.
303 */
304 private final BracketFinder bracket = new BracketFinder();
305
306 /**
307 * The "BrentOptimizer" default stopping criterion uses the tolerances
308 * to check the domain (point) values, not the function values.
309 * We thus create a custom checker to use function values.
310 *
311 * @param rel Relative threshold.
312 * @param abs Absolute threshold.
313 */
314 LineSearch(double rel,
315 double abs) {
316 super(REL_TOL_UNUSED,
317 ABS_TOL_UNUSED,
318 new SimpleUnivariateValueChecker(rel, abs));
319 }
320
321 /**
322 * Find the minimum of the function {@code f(p + alpha * d)}.
323 *
324 * @param p Starting point.
325 * @param d Search direction.
326 * @return the optimum.
327 * @throws org.apache.commons.math3.exception.TooManyEvaluationsException
328 * if the number of evaluations is exceeded.
329 */
330 public UnivariatePointValuePair search(final double[] p, final double[] d) {
331 final int n = p.length;
332 final UnivariateFunction f = new UnivariateFunction() {
333 public double value(double alpha) {
334 final double[] x = new double[n];
335 for (int i = 0; i < n; i++) {
336 x[i] = p[i] + alpha * d[i];
337 }
338 final double obj = PowellOptimizer.this.computeObjectiveValue(x);
339 return obj;
340 }
341 };
342
343 final GoalType goal = PowellOptimizer.this.getGoalType();
344 bracket.search(f, goal, 0, 1);
345 // Passing "MAX_VALUE" as a dummy value because it is the enclosing
346 // class that counts the number of evaluations (and will eventually
347 // generate the exception).
348 return optimize(new MaxEval(Integer.MAX_VALUE),
349 new UnivariateObjectiveFunction(f),
350 goal,
351 new SearchInterval(bracket.getLo(),
352 bracket.getHi(),
353 bracket.getMid()));
354 }
355 }
356 }