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    
018    package org.apache.commons.math3.optim.nonlinear.scalar.noderiv;
019    
020    import java.util.ArrayList;
021    import java.util.Arrays;
022    import java.util.List;
023    import org.apache.commons.math3.exception.DimensionMismatchException;
024    import org.apache.commons.math3.exception.NotPositiveException;
025    import org.apache.commons.math3.exception.NotStrictlyPositiveException;
026    import org.apache.commons.math3.exception.OutOfRangeException;
027    import org.apache.commons.math3.exception.TooManyEvaluationsException;
028    import org.apache.commons.math3.linear.Array2DRowRealMatrix;
029    import org.apache.commons.math3.linear.EigenDecomposition;
030    import org.apache.commons.math3.linear.MatrixUtils;
031    import org.apache.commons.math3.linear.RealMatrix;
032    import org.apache.commons.math3.optim.ConvergenceChecker;
033    import org.apache.commons.math3.optim.OptimizationData;
034    import org.apache.commons.math3.optim.nonlinear.scalar.GoalType;
035    import org.apache.commons.math3.optim.PointValuePair;
036    import org.apache.commons.math3.optim.nonlinear.scalar.MultivariateOptimizer;
037    import org.apache.commons.math3.random.RandomGenerator;
038    import org.apache.commons.math3.util.MathArrays;
039    
040    /**
041     * <p>An implementation of the active Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
042     * for non-linear, non-convex, non-smooth, global function minimization.
043     * The CMA-Evolution Strategy (CMA-ES) is a reliable stochastic optimization method
044     * which should be applied if derivative-based methods, e.g. quasi-Newton BFGS or
045     * conjugate gradient, fail due to a rugged search landscape (e.g. noise, local
046     * optima, outlier, etc.) of the objective function. Like a
047     * quasi-Newton method, the CMA-ES learns and applies a variable metric
048     * on the underlying search space. Unlike a quasi-Newton method, the
049     * CMA-ES neither estimates nor uses gradients, making it considerably more
050     * reliable in terms of finding a good, or even close to optimal, solution.</p>
051     *
052     * <p>In general, on smooth objective functions the CMA-ES is roughly ten times
053     * slower than BFGS (counting objective function evaluations, no gradients provided).
054     * For up to <math>N=10</math> variables also the derivative-free simplex
055     * direct search method (Nelder and Mead) can be faster, but it is
056     * far less reliable than CMA-ES.</p>
057     *
058     * <p>The CMA-ES is particularly well suited for non-separable
059     * and/or badly conditioned problems. To observe the advantage of CMA compared
060     * to a conventional evolution strategy, it will usually take about
061     * <math>30 N</math> function evaluations. On difficult problems the complete
062     * optimization (a single run) is expected to take <em>roughly</em> between
063     * <math>30 N</math> and <math>300 N<sup>2</sup></math>
064     * function evaluations.</p>
065     *
066     * <p>This implementation is translated and adapted from the Matlab version
067     * of the CMA-ES algorithm as implemented in module {@code cmaes.m} version 3.51.</p>
068     *
069     * For more information, please refer to the following links:
070     * <ul>
071     *  <li><a href="http://www.lri.fr/~hansen/cmaes.m">Matlab code</a></li>
072     *  <li><a href="http://www.lri.fr/~hansen/cmaesintro.html">Introduction to CMA-ES</a></li>
073     *  <li><a href="http://en.wikipedia.org/wiki/CMA-ES">Wikipedia</a></li>
074     * </ul>
075     *
076     * @version $Id: CMAESOptimizer.java 1462503 2013-03-29 15:48:27Z luc $
077     * @since 3.0
078     */
079    public class CMAESOptimizer
080        extends MultivariateOptimizer {
081        // global search parameters
082        /**
083         * Population size, offspring number. The primary strategy parameter to play
084         * with, which can be increased from its default value. Increasing the
085         * population size improves global search properties in exchange to speed.
086         * Speed decreases, as a rule, at most linearly with increasing population
087         * size. It is advisable to begin with the default small population size.
088         */
089        private int lambda; // population size
090        /**
091         * Covariance update mechanism, default is active CMA. isActiveCMA = true
092         * turns on "active CMA" with a negative update of the covariance matrix and
093         * checks for positive definiteness. OPTS.CMA.active = 2 does not check for
094         * pos. def. and is numerically faster. Active CMA usually speeds up the
095         * adaptation.
096         */
097        private final boolean isActiveCMA;
098        /**
099         * Determines how often a new random offspring is generated in case it is
100         * not feasible / beyond the defined limits, default is 0.
101         */
102        private final int checkFeasableCount;
103        /**
104         * @see Sigma
105         */
106        private double[] inputSigma;
107        /** Number of objective variables/problem dimension */
108        private int dimension;
109        /**
110         * Defines the number of initial iterations, where the covariance matrix
111         * remains diagonal and the algorithm has internally linear time complexity.
112         * diagonalOnly = 1 means keeping the covariance matrix always diagonal and
113         * this setting also exhibits linear space complexity. This can be
114         * particularly useful for dimension > 100.
115         * @see <a href="http://hal.archives-ouvertes.fr/inria-00287367/en">A Simple Modification in CMA-ES</a>
116         */
117        private int diagonalOnly;
118        /** Number of objective variables/problem dimension */
119        private boolean isMinimize = true;
120        /** Indicates whether statistic data is collected. */
121        private final boolean generateStatistics;
122    
123        // termination criteria
124        /** Maximal number of iterations allowed. */
125        private final int maxIterations;
126        /** Limit for fitness value. */
127        private final double stopFitness;
128        /** Stop if x-changes larger stopTolUpX. */
129        private double stopTolUpX;
130        /** Stop if x-change smaller stopTolX. */
131        private double stopTolX;
132        /** Stop if fun-changes smaller stopTolFun. */
133        private double stopTolFun;
134        /** Stop if back fun-changes smaller stopTolHistFun. */
135        private double stopTolHistFun;
136    
137        // selection strategy parameters
138        /** Number of parents/points for recombination. */
139        private int mu; //
140        /** log(mu + 0.5), stored for efficiency. */
141        private double logMu2;
142        /** Array for weighted recombination. */
143        private RealMatrix weights;
144        /** Variance-effectiveness of sum w_i x_i. */
145        private double mueff; //
146    
147        // dynamic strategy parameters and constants
148        /** Overall standard deviation - search volume. */
149        private double sigma;
150        /** Cumulation constant. */
151        private double cc;
152        /** Cumulation constant for step-size. */
153        private double cs;
154        /** Damping for step-size. */
155        private double damps;
156        /** Learning rate for rank-one update. */
157        private double ccov1;
158        /** Learning rate for rank-mu update' */
159        private double ccovmu;
160        /** Expectation of ||N(0,I)|| == norm(randn(N,1)). */
161        private double chiN;
162        /** Learning rate for rank-one update - diagonalOnly */
163        private double ccov1Sep;
164        /** Learning rate for rank-mu update - diagonalOnly */
165        private double ccovmuSep;
166    
167        // CMA internal values - updated each generation
168        /** Objective variables. */
169        private RealMatrix xmean;
170        /** Evolution path. */
171        private RealMatrix pc;
172        /** Evolution path for sigma. */
173        private RealMatrix ps;
174        /** Norm of ps, stored for efficiency. */
175        private double normps;
176        /** Coordinate system. */
177        private RealMatrix B;
178        /** Scaling. */
179        private RealMatrix D;
180        /** B*D, stored for efficiency. */
181        private RealMatrix BD;
182        /** Diagonal of sqrt(D), stored for efficiency. */
183        private RealMatrix diagD;
184        /** Covariance matrix. */
185        private RealMatrix C;
186        /** Diagonal of C, used for diagonalOnly. */
187        private RealMatrix diagC;
188        /** Number of iterations already performed. */
189        private int iterations;
190    
191        /** History queue of best values. */
192        private double[] fitnessHistory;
193        /** Size of history queue of best values. */
194        private int historySize;
195    
196        /** Random generator. */
197        private final RandomGenerator random;
198    
199        /** History of sigma values. */
200        private final List<Double> statisticsSigmaHistory = new ArrayList<Double>();
201        /** History of mean matrix. */
202        private final List<RealMatrix> statisticsMeanHistory = new ArrayList<RealMatrix>();
203        /** History of fitness values. */
204        private final List<Double> statisticsFitnessHistory = new ArrayList<Double>();
205        /** History of D matrix. */
206        private final List<RealMatrix> statisticsDHistory = new ArrayList<RealMatrix>();
207    
208        /**
209         * @param maxIterations Maximal number of iterations.
210         * @param stopFitness Whether to stop if objective function value is smaller than
211         * {@code stopFitness}.
212         * @param isActiveCMA Chooses the covariance matrix update method.
213         * @param diagonalOnly Number of initial iterations, where the covariance matrix
214         * remains diagonal.
215         * @param checkFeasableCount Determines how often new random objective variables are
216         * generated in case they are out of bounds.
217         * @param random Random generator.
218         * @param generateStatistics Whether statistic data is collected.
219         * @param checker Convergence checker.
220         *
221         * @since 3.1
222         */
223        public CMAESOptimizer(int maxIterations,
224                              double stopFitness,
225                              boolean isActiveCMA,
226                              int diagonalOnly,
227                              int checkFeasableCount,
228                              RandomGenerator random,
229                              boolean generateStatistics,
230                              ConvergenceChecker<PointValuePair> checker) {
231            super(checker);
232            this.maxIterations = maxIterations;
233            this.stopFitness = stopFitness;
234            this.isActiveCMA = isActiveCMA;
235            this.diagonalOnly = diagonalOnly;
236            this.checkFeasableCount = checkFeasableCount;
237            this.random = random;
238            this.generateStatistics = generateStatistics;
239        }
240    
241        /**
242         * @return History of sigma values.
243         */
244        public List<Double> getStatisticsSigmaHistory() {
245            return statisticsSigmaHistory;
246        }
247    
248        /**
249         * @return History of mean matrix.
250         */
251        public List<RealMatrix> getStatisticsMeanHistory() {
252            return statisticsMeanHistory;
253        }
254    
255        /**
256         * @return History of fitness values.
257         */
258        public List<Double> getStatisticsFitnessHistory() {
259            return statisticsFitnessHistory;
260        }
261    
262        /**
263         * @return History of D matrix.
264         */
265        public List<RealMatrix> getStatisticsDHistory() {
266            return statisticsDHistory;
267        }
268    
269        /**
270         * Input sigma values.
271         * They define the initial coordinate-wise standard deviations for
272         * sampling new search points around the initial guess.
273         * It is suggested to set them to the estimated distance from the
274         * initial to the desired optimum.
275         * Small values induce the search to be more local (and very small
276         * values are more likely to find a local optimum close to the initial
277         * guess).
278         * Too small values might however lead to early termination.
279         */
280        public static class Sigma implements OptimizationData {
281            /** Sigma values. */
282            private final double[] sigma;
283    
284            /**
285             * @param s Sigma values.
286             * @throws NotPositiveException if any of the array entries is smaller
287             * than zero.
288             */
289            public Sigma(double[] s)
290                throws NotPositiveException {
291                for (int i = 0; i < s.length; i++) {
292                    if (s[i] < 0) {
293                        throw new NotPositiveException(s[i]);
294                    }
295                }
296    
297                sigma = s.clone();
298            }
299    
300            /**
301             * @return the sigma values.
302             */
303            public double[] getSigma() {
304                return sigma.clone();
305            }
306        }
307    
308        /**
309         * Population size.
310         * The number of offspring is the primary strategy parameter.
311         * In the absence of better clues, a good default could be an
312         * integer close to {@code 4 + 3 ln(n)}, where {@code n} is the
313         * number of optimized parameters.
314         * Increasing the population size improves global search properties
315         * at the expense of speed (which in general decreases at most
316         * linearly with increasing population size).
317         */
318        public static class PopulationSize implements OptimizationData {
319            /** Population size. */
320            private final int lambda;
321    
322            /**
323             * @param size Population size.
324             * @throws NotStrictlyPositiveException if {@code size <= 0}.
325             */
326            public PopulationSize(int size)
327                throws NotStrictlyPositiveException {
328                if (size <= 0) {
329                    throw new NotStrictlyPositiveException(size);
330                }
331                lambda = size;
332            }
333    
334            /**
335             * @return the population size.
336             */
337            public int getPopulationSize() {
338                return lambda;
339            }
340        }
341    
342        /**
343         * {@inheritDoc}
344         *
345         * @param optData Optimization data. In addition to those documented in
346         * {@link MultivariateOptimizer#parseOptimizationData(OptimizationData[])
347         * MultivariateOptimizer}, this method will register the following data:
348         * <ul>
349         *  <li>{@link Sigma}</li>
350         *  <li>{@link PopulationSize}</li>
351         * </ul>
352         * @return {@inheritDoc}
353         * @throws TooManyEvaluationsException if the maximal number of
354         * evaluations is exceeded.
355         * @throws DimensionMismatchException if the initial guess, target, and weight
356         * arguments have inconsistent dimensions.
357         */
358        @Override
359        public PointValuePair optimize(OptimizationData... optData)
360            throws TooManyEvaluationsException,
361                   DimensionMismatchException {
362            // Set up base class and perform computation.
363            return super.optimize(optData);
364        }
365    
366        /** {@inheritDoc} */
367        @Override
368        protected PointValuePair doOptimize() {
369             // -------------------- Initialization --------------------------------
370            isMinimize = getGoalType().equals(GoalType.MINIMIZE);
371            final FitnessFunction fitfun = new FitnessFunction();
372            final double[] guess = getStartPoint();
373            // number of objective variables/problem dimension
374            dimension = guess.length;
375            initializeCMA(guess);
376            iterations = 0;
377            double bestValue = fitfun.value(guess);
378            push(fitnessHistory, bestValue);
379            PointValuePair optimum
380                = new PointValuePair(getStartPoint(),
381                                     isMinimize ? bestValue : -bestValue);
382            PointValuePair lastResult = null;
383    
384            // -------------------- Generation Loop --------------------------------
385    
386            generationLoop:
387            for (iterations = 1; iterations <= maxIterations; iterations++) {
388                incrementIterationCount();
389    
390                // Generate and evaluate lambda offspring
391                final RealMatrix arz = randn1(dimension, lambda);
392                final RealMatrix arx = zeros(dimension, lambda);
393                final double[] fitness = new double[lambda];
394                // generate random offspring
395                for (int k = 0; k < lambda; k++) {
396                    RealMatrix arxk = null;
397                    for (int i = 0; i < checkFeasableCount + 1; i++) {
398                        if (diagonalOnly <= 0) {
399                            arxk = xmean.add(BD.multiply(arz.getColumnMatrix(k))
400                                             .scalarMultiply(sigma)); // m + sig * Normal(0,C)
401                        } else {
402                            arxk = xmean.add(times(diagD,arz.getColumnMatrix(k))
403                                             .scalarMultiply(sigma));
404                        }
405                        if (i >= checkFeasableCount ||
406                            fitfun.isFeasible(arxk.getColumn(0))) {
407                            break;
408                        }
409                        // regenerate random arguments for row
410                        arz.setColumn(k, randn(dimension));
411                    }
412                    copyColumn(arxk, 0, arx, k);
413                    try {
414                        fitness[k] = fitfun.value(arx.getColumn(k)); // compute fitness
415                    } catch (TooManyEvaluationsException e) {
416                        break generationLoop;
417                    }
418                }
419                // Sort by fitness and compute weighted mean into xmean
420                final int[] arindex = sortedIndices(fitness);
421                // Calculate new xmean, this is selection and recombination
422                final RealMatrix xold = xmean; // for speed up of Eq. (2) and (3)
423                final RealMatrix bestArx = selectColumns(arx, MathArrays.copyOf(arindex, mu));
424                xmean = bestArx.multiply(weights);
425                final RealMatrix bestArz = selectColumns(arz, MathArrays.copyOf(arindex, mu));
426                final RealMatrix zmean = bestArz.multiply(weights);
427                final boolean hsig = updateEvolutionPaths(zmean, xold);
428                if (diagonalOnly <= 0) {
429                    updateCovariance(hsig, bestArx, arz, arindex, xold);
430                } else {
431                    updateCovarianceDiagonalOnly(hsig, bestArz);
432                }
433                // Adapt step size sigma - Eq. (5)
434                sigma *= Math.exp(Math.min(1, (normps/chiN - 1) * cs / damps));
435                final double bestFitness = fitness[arindex[0]];
436                final double worstFitness = fitness[arindex[arindex.length - 1]];
437                if (bestValue > bestFitness) {
438                    bestValue = bestFitness;
439                    lastResult = optimum;
440                    optimum = new PointValuePair(fitfun.repair(bestArx.getColumn(0)),
441                                                 isMinimize ? bestFitness : -bestFitness);
442                    if (getConvergenceChecker() != null && lastResult != null &&
443                        getConvergenceChecker().converged(iterations, optimum, lastResult)) {
444                        break generationLoop;
445                    }
446                }
447                // handle termination criteria
448                // Break, if fitness is good enough
449                if (stopFitness != 0 && bestFitness < (isMinimize ? stopFitness : -stopFitness)) {
450                    break generationLoop;
451                }
452                final double[] sqrtDiagC = sqrt(diagC).getColumn(0);
453                final double[] pcCol = pc.getColumn(0);
454                for (int i = 0; i < dimension; i++) {
455                    if (sigma * Math.max(Math.abs(pcCol[i]), sqrtDiagC[i]) > stopTolX) {
456                        break;
457                    }
458                    if (i >= dimension - 1) {
459                        break generationLoop;
460                    }
461                }
462                for (int i = 0; i < dimension; i++) {
463                    if (sigma * sqrtDiagC[i] > stopTolUpX) {
464                        break generationLoop;
465                    }
466                }
467                final double historyBest = min(fitnessHistory);
468                final double historyWorst = max(fitnessHistory);
469                if (iterations > 2 &&
470                    Math.max(historyWorst, worstFitness) -
471                    Math.min(historyBest, bestFitness) < stopTolFun) {
472                    break generationLoop;
473                }
474                if (iterations > fitnessHistory.length &&
475                    historyWorst - historyBest < stopTolHistFun) {
476                    break generationLoop;
477                }
478                // condition number of the covariance matrix exceeds 1e14
479                if (max(diagD) / min(diagD) > 1e7) {
480                    break generationLoop;
481                }
482                // user defined termination
483                if (getConvergenceChecker() != null) {
484                    final PointValuePair current
485                        = new PointValuePair(bestArx.getColumn(0),
486                                             isMinimize ? bestFitness : -bestFitness);
487                    if (lastResult != null &&
488                        getConvergenceChecker().converged(iterations, current, lastResult)) {
489                        break generationLoop;
490                        }
491                    lastResult = current;
492                }
493                // Adjust step size in case of equal function values (flat fitness)
494                if (bestValue == fitness[arindex[(int)(0.1+lambda/4.)]]) {
495                    sigma = sigma * Math.exp(0.2 + cs / damps);
496                }
497                if (iterations > 2 && Math.max(historyWorst, bestFitness) -
498                    Math.min(historyBest, bestFitness) == 0) {
499                    sigma = sigma * Math.exp(0.2 + cs / damps);
500                }
501                // store best in history
502                push(fitnessHistory,bestFitness);
503                fitfun.setValueRange(worstFitness-bestFitness);
504                if (generateStatistics) {
505                    statisticsSigmaHistory.add(sigma);
506                    statisticsFitnessHistory.add(bestFitness);
507                    statisticsMeanHistory.add(xmean.transpose());
508                    statisticsDHistory.add(diagD.transpose().scalarMultiply(1E5));
509                }
510            }
511            return optimum;
512        }
513    
514        /**
515         * Scans the list of (required and optional) optimization data that
516         * characterize the problem.
517         *
518         * @param optData Optimization data. The following data will be looked for:
519         * <ul>
520         *  <li>{@link Sigma}</li>
521         *  <li>{@link PopulationSize}</li>
522         * </ul>
523         */
524        @Override
525        protected void parseOptimizationData(OptimizationData... optData) {
526            // Allow base class to register its own data.
527            super.parseOptimizationData(optData);
528    
529            // The existing values (as set by the previous call) are reused if
530            // not provided in the argument list.
531            for (OptimizationData data : optData) {
532                if (data instanceof Sigma) {
533                    inputSigma = ((Sigma) data).getSigma();
534                    continue;
535                }
536                if (data instanceof PopulationSize) {
537                    lambda = ((PopulationSize) data).getPopulationSize();
538                    continue;
539                }
540            }
541    
542            checkParameters();
543        }
544    
545        /**
546         * Checks dimensions and values of boundaries and inputSigma if defined.
547         */
548        private void checkParameters() {
549            final double[] init = getStartPoint();
550            final double[] lB = getLowerBound();
551            final double[] uB = getUpperBound();
552    
553            if (inputSigma != null) {
554                if (inputSigma.length != init.length) {
555                    throw new DimensionMismatchException(inputSigma.length, init.length);
556                }
557                for (int i = 0; i < init.length; i++) {
558                    if (inputSigma[i] > uB[i] - lB[i]) {
559                        throw new OutOfRangeException(inputSigma[i], 0, uB[i] - lB[i]);
560                    }
561                }
562            }
563        }
564    
565        /**
566         * Initialization of the dynamic search parameters
567         *
568         * @param guess Initial guess for the arguments of the fitness function.
569         */
570        private void initializeCMA(double[] guess) {
571            if (lambda <= 0) {
572                throw new NotStrictlyPositiveException(lambda);
573            }
574            // initialize sigma
575            final double[][] sigmaArray = new double[guess.length][1];
576            for (int i = 0; i < guess.length; i++) {
577                sigmaArray[i][0] = inputSigma[i];
578            }
579            final RealMatrix insigma = new Array2DRowRealMatrix(sigmaArray, false);
580            sigma = max(insigma); // overall standard deviation
581    
582            // initialize termination criteria
583            stopTolUpX = 1e3 * max(insigma);
584            stopTolX = 1e-11 * max(insigma);
585            stopTolFun = 1e-12;
586            stopTolHistFun = 1e-13;
587    
588            // initialize selection strategy parameters
589            mu = lambda / 2; // number of parents/points for recombination
590            logMu2 = Math.log(mu + 0.5);
591            weights = log(sequence(1, mu, 1)).scalarMultiply(-1).scalarAdd(logMu2);
592            double sumw = 0;
593            double sumwq = 0;
594            for (int i = 0; i < mu; i++) {
595                double w = weights.getEntry(i, 0);
596                sumw += w;
597                sumwq += w * w;
598            }
599            weights = weights.scalarMultiply(1 / sumw);
600            mueff = sumw * sumw / sumwq; // variance-effectiveness of sum w_i x_i
601    
602            // initialize dynamic strategy parameters and constants
603            cc = (4 + mueff / dimension) /
604                    (dimension + 4 + 2 * mueff / dimension);
605            cs = (mueff + 2) / (dimension + mueff + 3.);
606            damps = (1 + 2 * Math.max(0, Math.sqrt((mueff - 1) /
607                                                   (dimension + 1)) - 1)) *
608                Math.max(0.3,
609                         1 - dimension / (1e-6 + maxIterations)) + cs; // minor increment
610            ccov1 = 2 / ((dimension + 1.3) * (dimension + 1.3) + mueff);
611            ccovmu = Math.min(1 - ccov1, 2 * (mueff - 2 + 1 / mueff) /
612                              ((dimension + 2) * (dimension + 2) + mueff));
613            ccov1Sep = Math.min(1, ccov1 * (dimension + 1.5) / 3);
614            ccovmuSep = Math.min(1 - ccov1, ccovmu * (dimension + 1.5) / 3);
615            chiN = Math.sqrt(dimension) *
616                (1 - 1 / ((double) 4 * dimension) + 1 / ((double) 21 * dimension * dimension));
617            // intialize CMA internal values - updated each generation
618            xmean = MatrixUtils.createColumnRealMatrix(guess); // objective variables
619            diagD = insigma.scalarMultiply(1 / sigma);
620            diagC = square(diagD);
621            pc = zeros(dimension, 1); // evolution paths for C and sigma
622            ps = zeros(dimension, 1); // B defines the coordinate system
623            normps = ps.getFrobeniusNorm();
624    
625            B = eye(dimension, dimension);
626            D = ones(dimension, 1); // diagonal D defines the scaling
627            BD = times(B, repmat(diagD.transpose(), dimension, 1));
628            C = B.multiply(diag(square(D)).multiply(B.transpose())); // covariance
629            historySize = 10 + (int) (3 * 10 * dimension / (double) lambda);
630            fitnessHistory = new double[historySize]; // history of fitness values
631            for (int i = 0; i < historySize; i++) {
632                fitnessHistory[i] = Double.MAX_VALUE;
633            }
634        }
635    
636        /**
637         * Update of the evolution paths ps and pc.
638         *
639         * @param zmean Weighted row matrix of the gaussian random numbers generating
640         * the current offspring.
641         * @param xold xmean matrix of the previous generation.
642         * @return hsig flag indicating a small correction.
643         */
644        private boolean updateEvolutionPaths(RealMatrix zmean, RealMatrix xold) {
645            ps = ps.scalarMultiply(1 - cs).add(
646                    B.multiply(zmean).scalarMultiply(
647                            Math.sqrt(cs * (2 - cs) * mueff)));
648            normps = ps.getFrobeniusNorm();
649            final boolean hsig = normps /
650                Math.sqrt(1 - Math.pow(1 - cs, 2 * iterations)) /
651                chiN < 1.4 + 2 / ((double) dimension + 1);
652            pc = pc.scalarMultiply(1 - cc);
653            if (hsig) {
654                pc = pc.add(xmean.subtract(xold).scalarMultiply(Math.sqrt(cc * (2 - cc) * mueff) / sigma));
655            }
656            return hsig;
657        }
658    
659        /**
660         * Update of the covariance matrix C for diagonalOnly > 0
661         *
662         * @param hsig Flag indicating a small correction.
663         * @param bestArz Fitness-sorted matrix of the gaussian random values of the
664         * current offspring.
665         */
666        private void updateCovarianceDiagonalOnly(boolean hsig,
667                                                  final RealMatrix bestArz) {
668            // minor correction if hsig==false
669            double oldFac = hsig ? 0 : ccov1Sep * cc * (2 - cc);
670            oldFac += 1 - ccov1Sep - ccovmuSep;
671            diagC = diagC.scalarMultiply(oldFac) // regard old matrix
672                .add(square(pc).scalarMultiply(ccov1Sep)) // plus rank one update
673                .add((times(diagC, square(bestArz).multiply(weights))) // plus rank mu update
674                     .scalarMultiply(ccovmuSep));
675            diagD = sqrt(diagC); // replaces eig(C)
676            if (diagonalOnly > 1 &&
677                iterations > diagonalOnly) {
678                // full covariance matrix from now on
679                diagonalOnly = 0;
680                B = eye(dimension, dimension);
681                BD = diag(diagD);
682                C = diag(diagC);
683            }
684        }
685    
686        /**
687         * Update of the covariance matrix C.
688         *
689         * @param hsig Flag indicating a small correction.
690         * @param bestArx Fitness-sorted matrix of the argument vectors producing the
691         * current offspring.
692         * @param arz Unsorted matrix containing the gaussian random values of the
693         * current offspring.
694         * @param arindex Indices indicating the fitness-order of the current offspring.
695         * @param xold xmean matrix of the previous generation.
696         */
697        private void updateCovariance(boolean hsig, final RealMatrix bestArx,
698                                      final RealMatrix arz, final int[] arindex,
699                                      final RealMatrix xold) {
700            double negccov = 0;
701            if (ccov1 + ccovmu > 0) {
702                final RealMatrix arpos = bestArx.subtract(repmat(xold, 1, mu))
703                    .scalarMultiply(1 / sigma); // mu difference vectors
704                final RealMatrix roneu = pc.multiply(pc.transpose())
705                    .scalarMultiply(ccov1); // rank one update
706                // minor correction if hsig==false
707                double oldFac = hsig ? 0 : ccov1 * cc * (2 - cc);
708                oldFac += 1 - ccov1 - ccovmu;
709                if (isActiveCMA) {
710                    // Adapt covariance matrix C active CMA
711                    negccov = (1 - ccovmu) * 0.25 * mueff /
712                        (Math.pow(dimension + 2, 1.5) + 2 * mueff);
713                    // keep at least 0.66 in all directions, small popsize are most
714                    // critical
715                    final double negminresidualvariance = 0.66;
716                    // where to make up for the variance loss
717                    final double negalphaold = 0.5;
718                    // prepare vectors, compute negative updating matrix Cneg
719                    final int[] arReverseIndex = reverse(arindex);
720                    RealMatrix arzneg = selectColumns(arz, MathArrays.copyOf(arReverseIndex, mu));
721                    RealMatrix arnorms = sqrt(sumRows(square(arzneg)));
722                    final int[] idxnorms = sortedIndices(arnorms.getRow(0));
723                    final RealMatrix arnormsSorted = selectColumns(arnorms, idxnorms);
724                    final int[] idxReverse = reverse(idxnorms);
725                    final RealMatrix arnormsReverse = selectColumns(arnorms, idxReverse);
726                    arnorms = divide(arnormsReverse, arnormsSorted);
727                    final int[] idxInv = inverse(idxnorms);
728                    final RealMatrix arnormsInv = selectColumns(arnorms, idxInv);
729                    // check and set learning rate negccov
730                    final double negcovMax = (1 - negminresidualvariance) /
731                        square(arnormsInv).multiply(weights).getEntry(0, 0);
732                    if (negccov > negcovMax) {
733                        negccov = negcovMax;
734                    }
735                    arzneg = times(arzneg, repmat(arnormsInv, dimension, 1));
736                    final RealMatrix artmp = BD.multiply(arzneg);
737                    final RealMatrix Cneg = artmp.multiply(diag(weights)).multiply(artmp.transpose());
738                    oldFac += negalphaold * negccov;
739                    C = C.scalarMultiply(oldFac)
740                        .add(roneu) // regard old matrix
741                        .add(arpos.scalarMultiply( // plus rank one update
742                                                  ccovmu + (1 - negalphaold) * negccov) // plus rank mu update
743                             .multiply(times(repmat(weights, 1, dimension),
744                                             arpos.transpose())))
745                        .subtract(Cneg.scalarMultiply(negccov));
746                } else {
747                    // Adapt covariance matrix C - nonactive
748                    C = C.scalarMultiply(oldFac) // regard old matrix
749                        .add(roneu) // plus rank one update
750                        .add(arpos.scalarMultiply(ccovmu) // plus rank mu update
751                             .multiply(times(repmat(weights, 1, dimension),
752                                             arpos.transpose())));
753                }
754            }
755            updateBD(negccov);
756        }
757    
758        /**
759         * Update B and D from C.
760         *
761         * @param negccov Negative covariance factor.
762         */
763        private void updateBD(double negccov) {
764            if (ccov1 + ccovmu + negccov > 0 &&
765                (iterations % 1. / (ccov1 + ccovmu + negccov) / dimension / 10.) < 1) {
766                // to achieve O(N^2)
767                C = triu(C, 0).add(triu(C, 1).transpose());
768                // enforce symmetry to prevent complex numbers
769                final EigenDecomposition eig = new EigenDecomposition(C);
770                B = eig.getV(); // eigen decomposition, B==normalized eigenvectors
771                D = eig.getD();
772                diagD = diag(D);
773                if (min(diagD) <= 0) {
774                    for (int i = 0; i < dimension; i++) {
775                        if (diagD.getEntry(i, 0) < 0) {
776                            diagD.setEntry(i, 0, 0);
777                        }
778                    }
779                    final double tfac = max(diagD) / 1e14;
780                    C = C.add(eye(dimension, dimension).scalarMultiply(tfac));
781                    diagD = diagD.add(ones(dimension, 1).scalarMultiply(tfac));
782                }
783                if (max(diagD) > 1e14 * min(diagD)) {
784                    final double tfac = max(diagD) / 1e14 - min(diagD);
785                    C = C.add(eye(dimension, dimension).scalarMultiply(tfac));
786                    diagD = diagD.add(ones(dimension, 1).scalarMultiply(tfac));
787                }
788                diagC = diag(C);
789                diagD = sqrt(diagD); // D contains standard deviations now
790                BD = times(B, repmat(diagD.transpose(), dimension, 1)); // O(n^2)
791            }
792        }
793    
794        /**
795         * Pushes the current best fitness value in a history queue.
796         *
797         * @param vals History queue.
798         * @param val Current best fitness value.
799         */
800        private static void push(double[] vals, double val) {
801            for (int i = vals.length-1; i > 0; i--) {
802                vals[i] = vals[i-1];
803            }
804            vals[0] = val;
805        }
806    
807        /**
808         * Sorts fitness values.
809         *
810         * @param doubles Array of values to be sorted.
811         * @return a sorted array of indices pointing into doubles.
812         */
813        private int[] sortedIndices(final double[] doubles) {
814            final DoubleIndex[] dis = new DoubleIndex[doubles.length];
815            for (int i = 0; i < doubles.length; i++) {
816                dis[i] = new DoubleIndex(doubles[i], i);
817            }
818            Arrays.sort(dis);
819            final int[] indices = new int[doubles.length];
820            for (int i = 0; i < doubles.length; i++) {
821                indices[i] = dis[i].index;
822            }
823            return indices;
824        }
825    
826        /**
827         * Used to sort fitness values. Sorting is always in lower value first
828         * order.
829         */
830        private static class DoubleIndex implements Comparable<DoubleIndex> {
831            /** Value to compare. */
832            private final double value;
833            /** Index into sorted array. */
834            private final int index;
835    
836            /**
837             * @param value Value to compare.
838             * @param index Index into sorted array.
839             */
840            DoubleIndex(double value, int index) {
841                this.value = value;
842                this.index = index;
843            }
844    
845            /** {@inheritDoc} */
846            public int compareTo(DoubleIndex o) {
847                return Double.compare(value, o.value);
848            }
849    
850            /** {@inheritDoc} */
851            @Override
852            public boolean equals(Object other) {
853    
854                if (this == other) {
855                    return true;
856                }
857    
858                if (other instanceof DoubleIndex) {
859                    return Double.compare(value, ((DoubleIndex) other).value) == 0;
860                }
861    
862                return false;
863            }
864    
865            /** {@inheritDoc} */
866            @Override
867            public int hashCode() {
868                long bits = Double.doubleToLongBits(value);
869                return (int) ((1438542 ^ (bits >>> 32) ^ bits) & 0xffffffff);
870            }
871        }
872    
873        /**
874         * Normalizes fitness values to the range [0,1]. Adds a penalty to the
875         * fitness value if out of range. The penalty is adjusted by calling
876         * setValueRange().
877         */
878        private class FitnessFunction {
879            /** Determines the penalty for boundary violations */
880            private double valueRange;
881            /**
882             * Flag indicating whether the objective variables are forced into their
883             * bounds if defined
884             */
885            private final boolean isRepairMode;
886    
887            /** Simple constructor.
888             */
889            public FitnessFunction() {
890                valueRange = 1;
891                isRepairMode = true;
892            }
893    
894            /**
895             * @param point Normalized objective variables.
896             * @return the objective value + penalty for violated bounds.
897             */
898            public double value(final double[] point) {
899                double value;
900                if (isRepairMode) {
901                    double[] repaired = repair(point);
902                    value = CMAESOptimizer.this.computeObjectiveValue(repaired) +
903                        penalty(point, repaired);
904                } else {
905                    value = CMAESOptimizer.this.computeObjectiveValue(point);
906                }
907                return isMinimize ? value : -value;
908            }
909    
910            /**
911             * @param x Normalized objective variables.
912             * @return {@code true} if in bounds.
913             */
914            public boolean isFeasible(final double[] x) {
915                final double[] lB = CMAESOptimizer.this.getLowerBound();
916                final double[] uB = CMAESOptimizer.this.getUpperBound();
917    
918                for (int i = 0; i < x.length; i++) {
919                    if (x[i] < lB[i]) {
920                        return false;
921                    }
922                    if (x[i] > uB[i]) {
923                        return false;
924                    }
925                }
926                return true;
927            }
928    
929            /**
930             * @param valueRange Adjusts the penalty computation.
931             */
932            public void setValueRange(double valueRange) {
933                this.valueRange = valueRange;
934            }
935    
936            /**
937             * @param x Normalized objective variables.
938             * @return the repaired (i.e. all in bounds) objective variables.
939             */
940            private double[] repair(final double[] x) {
941                final double[] lB = CMAESOptimizer.this.getLowerBound();
942                final double[] uB = CMAESOptimizer.this.getUpperBound();
943    
944                final double[] repaired = new double[x.length];
945                for (int i = 0; i < x.length; i++) {
946                    if (x[i] < lB[i]) {
947                        repaired[i] = lB[i];
948                    } else if (x[i] > uB[i]) {
949                        repaired[i] = uB[i];
950                    } else {
951                        repaired[i] = x[i];
952                    }
953                }
954                return repaired;
955            }
956    
957            /**
958             * @param x Normalized objective variables.
959             * @param repaired Repaired objective variables.
960             * @return Penalty value according to the violation of the bounds.
961             */
962            private double penalty(final double[] x, final double[] repaired) {
963                double penalty = 0;
964                for (int i = 0; i < x.length; i++) {
965                    double diff = Math.abs(x[i] - repaired[i]);
966                    penalty += diff * valueRange;
967                }
968                return isMinimize ? penalty : -penalty;
969            }
970        }
971    
972        // -----Matrix utility functions similar to the Matlab build in functions------
973    
974        /**
975         * @param m Input matrix
976         * @return Matrix representing the element-wise logarithm of m.
977         */
978        private static RealMatrix log(final RealMatrix m) {
979            final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
980            for (int r = 0; r < m.getRowDimension(); r++) {
981                for (int c = 0; c < m.getColumnDimension(); c++) {
982                    d[r][c] = Math.log(m.getEntry(r, c));
983                }
984            }
985            return new Array2DRowRealMatrix(d, false);
986        }
987    
988        /**
989         * @param m Input matrix.
990         * @return Matrix representing the element-wise square root of m.
991         */
992        private static RealMatrix sqrt(final RealMatrix m) {
993            final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
994            for (int r = 0; r < m.getRowDimension(); r++) {
995                for (int c = 0; c < m.getColumnDimension(); c++) {
996                    d[r][c] = Math.sqrt(m.getEntry(r, c));
997                }
998            }
999            return new Array2DRowRealMatrix(d, false);
1000        }
1001    
1002        /**
1003         * @param m Input matrix.
1004         * @return Matrix representing the element-wise square of m.
1005         */
1006        private static RealMatrix square(final RealMatrix m) {
1007            final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
1008            for (int r = 0; r < m.getRowDimension(); r++) {
1009                for (int c = 0; c < m.getColumnDimension(); c++) {
1010                    double e = m.getEntry(r, c);
1011                    d[r][c] = e * e;
1012                }
1013            }
1014            return new Array2DRowRealMatrix(d, false);
1015        }
1016    
1017        /**
1018         * @param m Input matrix 1.
1019         * @param n Input matrix 2.
1020         * @return the matrix where the elements of m and n are element-wise multiplied.
1021         */
1022        private static RealMatrix times(final RealMatrix m, final RealMatrix n) {
1023            final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
1024            for (int r = 0; r < m.getRowDimension(); r++) {
1025                for (int c = 0; c < m.getColumnDimension(); c++) {
1026                    d[r][c] = m.getEntry(r, c) * n.getEntry(r, c);
1027                }
1028            }
1029            return new Array2DRowRealMatrix(d, false);
1030        }
1031    
1032        /**
1033         * @param m Input matrix 1.
1034         * @param n Input matrix 2.
1035         * @return Matrix where the elements of m and n are element-wise divided.
1036         */
1037        private static RealMatrix divide(final RealMatrix m, final RealMatrix n) {
1038            final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
1039            for (int r = 0; r < m.getRowDimension(); r++) {
1040                for (int c = 0; c < m.getColumnDimension(); c++) {
1041                    d[r][c] = m.getEntry(r, c) / n.getEntry(r, c);
1042                }
1043            }
1044            return new Array2DRowRealMatrix(d, false);
1045        }
1046    
1047        /**
1048         * @param m Input matrix.
1049         * @param cols Columns to select.
1050         * @return Matrix representing the selected columns.
1051         */
1052        private static RealMatrix selectColumns(final RealMatrix m, final int[] cols) {
1053            final double[][] d = new double[m.getRowDimension()][cols.length];
1054            for (int r = 0; r < m.getRowDimension(); r++) {
1055                for (int c = 0; c < cols.length; c++) {
1056                    d[r][c] = m.getEntry(r, cols[c]);
1057                }
1058            }
1059            return new Array2DRowRealMatrix(d, false);
1060        }
1061    
1062        /**
1063         * @param m Input matrix.
1064         * @param k Diagonal position.
1065         * @return Upper triangular part of matrix.
1066         */
1067        private static RealMatrix triu(final RealMatrix m, int k) {
1068            final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
1069            for (int r = 0; r < m.getRowDimension(); r++) {
1070                for (int c = 0; c < m.getColumnDimension(); c++) {
1071                    d[r][c] = r <= c - k ? m.getEntry(r, c) : 0;
1072                }
1073            }
1074            return new Array2DRowRealMatrix(d, false);
1075        }
1076    
1077        /**
1078         * @param m Input matrix.
1079         * @return Row matrix representing the sums of the rows.
1080         */
1081        private static RealMatrix sumRows(final RealMatrix m) {
1082            final double[][] d = new double[1][m.getColumnDimension()];
1083            for (int c = 0; c < m.getColumnDimension(); c++) {
1084                double sum = 0;
1085                for (int r = 0; r < m.getRowDimension(); r++) {
1086                    sum += m.getEntry(r, c);
1087                }
1088                d[0][c] = sum;
1089            }
1090            return new Array2DRowRealMatrix(d, false);
1091        }
1092    
1093        /**
1094         * @param m Input matrix.
1095         * @return the diagonal n-by-n matrix if m is a column matrix or the column
1096         * matrix representing the diagonal if m is a n-by-n matrix.
1097         */
1098        private static RealMatrix diag(final RealMatrix m) {
1099            if (m.getColumnDimension() == 1) {
1100                final double[][] d = new double[m.getRowDimension()][m.getRowDimension()];
1101                for (int i = 0; i < m.getRowDimension(); i++) {
1102                    d[i][i] = m.getEntry(i, 0);
1103                }
1104                return new Array2DRowRealMatrix(d, false);
1105            } else {
1106                final double[][] d = new double[m.getRowDimension()][1];
1107                for (int i = 0; i < m.getColumnDimension(); i++) {
1108                    d[i][0] = m.getEntry(i, i);
1109                }
1110                return new Array2DRowRealMatrix(d, false);
1111            }
1112        }
1113    
1114        /**
1115         * Copies a column from m1 to m2.
1116         *
1117         * @param m1 Source matrix.
1118         * @param col1 Source column.
1119         * @param m2 Target matrix.
1120         * @param col2 Target column.
1121         */
1122        private static void copyColumn(final RealMatrix m1, int col1,
1123                                       RealMatrix m2, int col2) {
1124            for (int i = 0; i < m1.getRowDimension(); i++) {
1125                m2.setEntry(i, col2, m1.getEntry(i, col1));
1126            }
1127        }
1128    
1129        /**
1130         * @param n Number of rows.
1131         * @param m Number of columns.
1132         * @return n-by-m matrix filled with 1.
1133         */
1134        private static RealMatrix ones(int n, int m) {
1135            final double[][] d = new double[n][m];
1136            for (int r = 0; r < n; r++) {
1137                Arrays.fill(d[r], 1);
1138            }
1139            return new Array2DRowRealMatrix(d, false);
1140        }
1141    
1142        /**
1143         * @param n Number of rows.
1144         * @param m Number of columns.
1145         * @return n-by-m matrix of 0 values out of diagonal, and 1 values on
1146         * the diagonal.
1147         */
1148        private static RealMatrix eye(int n, int m) {
1149            final double[][] d = new double[n][m];
1150            for (int r = 0; r < n; r++) {
1151                if (r < m) {
1152                    d[r][r] = 1;
1153                }
1154            }
1155            return new Array2DRowRealMatrix(d, false);
1156        }
1157    
1158        /**
1159         * @param n Number of rows.
1160         * @param m Number of columns.
1161         * @return n-by-m matrix of zero values.
1162         */
1163        private static RealMatrix zeros(int n, int m) {
1164            return new Array2DRowRealMatrix(n, m);
1165        }
1166    
1167        /**
1168         * @param mat Input matrix.
1169         * @param n Number of row replicates.
1170         * @param m Number of column replicates.
1171         * @return a matrix which replicates the input matrix in both directions.
1172         */
1173        private static RealMatrix repmat(final RealMatrix mat, int n, int m) {
1174            final int rd = mat.getRowDimension();
1175            final int cd = mat.getColumnDimension();
1176            final double[][] d = new double[n * rd][m * cd];
1177            for (int r = 0; r < n * rd; r++) {
1178                for (int c = 0; c < m * cd; c++) {
1179                    d[r][c] = mat.getEntry(r % rd, c % cd);
1180                }
1181            }
1182            return new Array2DRowRealMatrix(d, false);
1183        }
1184    
1185        /**
1186         * @param start Start value.
1187         * @param end End value.
1188         * @param step Step size.
1189         * @return a sequence as column matrix.
1190         */
1191        private static RealMatrix sequence(double start, double end, double step) {
1192            final int size = (int) ((end - start) / step + 1);
1193            final double[][] d = new double[size][1];
1194            double value = start;
1195            for (int r = 0; r < size; r++) {
1196                d[r][0] = value;
1197                value += step;
1198            }
1199            return new Array2DRowRealMatrix(d, false);
1200        }
1201    
1202        /**
1203         * @param m Input matrix.
1204         * @return the maximum of the matrix element values.
1205         */
1206        private static double max(final RealMatrix m) {
1207            double max = -Double.MAX_VALUE;
1208            for (int r = 0; r < m.getRowDimension(); r++) {
1209                for (int c = 0; c < m.getColumnDimension(); c++) {
1210                    double e = m.getEntry(r, c);
1211                    if (max < e) {
1212                        max = e;
1213                    }
1214                }
1215            }
1216            return max;
1217        }
1218    
1219        /**
1220         * @param m Input matrix.
1221         * @return the minimum of the matrix element values.
1222         */
1223        private static double min(final RealMatrix m) {
1224            double min = Double.MAX_VALUE;
1225            for (int r = 0; r < m.getRowDimension(); r++) {
1226                for (int c = 0; c < m.getColumnDimension(); c++) {
1227                    double e = m.getEntry(r, c);
1228                    if (min > e) {
1229                        min = e;
1230                    }
1231                }
1232            }
1233            return min;
1234        }
1235    
1236        /**
1237         * @param m Input array.
1238         * @return the maximum of the array values.
1239         */
1240        private static double max(final double[] m) {
1241            double max = -Double.MAX_VALUE;
1242            for (int r = 0; r < m.length; r++) {
1243                if (max < m[r]) {
1244                    max = m[r];
1245                }
1246            }
1247            return max;
1248        }
1249    
1250        /**
1251         * @param m Input array.
1252         * @return the minimum of the array values.
1253         */
1254        private static double min(final double[] m) {
1255            double min = Double.MAX_VALUE;
1256            for (int r = 0; r < m.length; r++) {
1257                if (min > m[r]) {
1258                    min = m[r];
1259                }
1260            }
1261            return min;
1262        }
1263    
1264        /**
1265         * @param indices Input index array.
1266         * @return the inverse of the mapping defined by indices.
1267         */
1268        private static int[] inverse(final int[] indices) {
1269            final int[] inverse = new int[indices.length];
1270            for (int i = 0; i < indices.length; i++) {
1271                inverse[indices[i]] = i;
1272            }
1273            return inverse;
1274        }
1275    
1276        /**
1277         * @param indices Input index array.
1278         * @return the indices in inverse order (last is first).
1279         */
1280        private static int[] reverse(final int[] indices) {
1281            final int[] reverse = new int[indices.length];
1282            for (int i = 0; i < indices.length; i++) {
1283                reverse[i] = indices[indices.length - i - 1];
1284            }
1285            return reverse;
1286        }
1287    
1288        /**
1289         * @param size Length of random array.
1290         * @return an array of Gaussian random numbers.
1291         */
1292        private double[] randn(int size) {
1293            final double[] randn = new double[size];
1294            for (int i = 0; i < size; i++) {
1295                randn[i] = random.nextGaussian();
1296            }
1297            return randn;
1298        }
1299    
1300        /**
1301         * @param size Number of rows.
1302         * @param popSize Population size.
1303         * @return a 2-dimensional matrix of Gaussian random numbers.
1304         */
1305        private RealMatrix randn1(int size, int popSize) {
1306            final double[][] d = new double[size][popSize];
1307            for (int r = 0; r < size; r++) {
1308                for (int c = 0; c < popSize; c++) {
1309                    d[r][c] = random.nextGaussian();
1310                }
1311            }
1312            return new Array2DRowRealMatrix(d, false);
1313        }
1314    }