CMAESOptimizer.java

  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.  *      https://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.  */

  18. /*
  19.  * This is not the original file distributed by the Apache Software Foundation
  20.  * It has been modified by the Hipparchus project
  21.  */

  23. package org.hipparchus.optim.nonlinear.scalar.noderiv;

  25. import java.util.ArrayList;
  26. import java.util.Arrays;
  27. import java.util.List;

  29. import org.hipparchus.exception.LocalizedCoreFormats;
  30. import org.hipparchus.exception.MathIllegalArgumentException;
  31. import org.hipparchus.exception.MathIllegalStateException;
  32. import org.hipparchus.linear.Array2DRowRealMatrix;
  33. import org.hipparchus.linear.EigenDecompositionSymmetric;
  34. import org.hipparchus.linear.MatrixUtils;
  35. import org.hipparchus.linear.RealMatrix;
  36. import org.hipparchus.optim.ConvergenceChecker;
  37. import org.hipparchus.optim.OptimizationData;
  38. import org.hipparchus.optim.PointValuePair;
  39. import org.hipparchus.optim.nonlinear.scalar.GoalType;
  40. import org.hipparchus.optim.nonlinear.scalar.MultivariateOptimizer;
  41. import org.hipparchus.random.RandomGenerator;
  42. import org.hipparchus.util.FastMath;

  44. /**
  45.  * An implementation of the active Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
  46.  * for non-linear, non-convex, non-smooth, global function minimization.
  47.  * <p>
  48.  * The CMA-Evolution Strategy (CMA-ES) is a reliable stochastic optimization method
  49.  * which should be applied if derivative-based methods, e.g. quasi-Newton BFGS or
  50.  * conjugate gradient, fail due to a rugged search landscape (e.g. noise, local
  51.  * optima, outlier, etc.) of the objective function. Like a
  52.  * quasi-Newton method, the CMA-ES learns and applies a variable metric
  53.  * on the underlying search space. Unlike a quasi-Newton method, the
  54.  * CMA-ES neither estimates nor uses gradients, making it considerably more
  55.  * reliable in terms of finding a good, or even close to optimal, solution.
  56.  * <p>
  57.  * In general, on smooth objective functions the CMA-ES is roughly ten times
  58.  * slower than BFGS (counting objective function evaluations, no gradients provided).
  59.  * For up to \(n=10\) variables also the derivative-free simplex
  60.  * direct search method (Nelder and Mead) can be faster, but it is
  61.  * far less reliable than CMA-ES.
  62.  * <p>
  63.  * The CMA-ES is particularly well suited for non-separable
  64.  * and/or badly conditioned problems. To observe the advantage of CMA compared
  65.  * to a conventional evolution strategy, it will usually take about
  66.  * \(30 n\) function evaluations. On difficult problems the complete
  67.  * optimization (a single run) is expected to take <em>roughly</em> between
  68.  * \(30 n\) and \(300 n^2\)
  69.  * function evaluations.
  70.  * <p>
  71.  * This implementation is translated and adapted from the Matlab version
  72.  * of the CMA-ES algorithm as implemented in module {@code cmaes.m} version 3.51.
  73.  * <p>
  74.  * For more information, please refer to the following links:
  75.  * <ul>
  76.  *  <li><a href="http://www.lri.fr/~hansen/cmaes.m">Matlab code</a></li>
  77.  *  <li><a href="http://www.lri.fr/~hansen/cmaesintro.html">Introduction to CMA-ES</a></li>
  78.  *  <li><a href="http://en.wikipedia.org/wiki/CMA-ES">Wikipedia</a></li>
  79.  * </ul>
  80.  *
  81.  */
  82. public class CMAESOptimizer
  83.     extends MultivariateOptimizer {
  84.     // global search parameters
  85.     /**
  86.      * Population size, offspring number. The primary strategy parameter to play
  87.      * with, which can be increased from its default value. Increasing the
  88.      * population size improves global search properties in exchange to speed.
  89.      * Speed decreases, as a rule, at most linearly with increasing population
  90.      * size. It is advisable to begin with the default small population size.
  91.      */
  92.     private int lambda; // population size
  93.     /**
  94.      * Covariance update mechanism, default is active CMA. isActiveCMA = true
  95.      * turns on "active CMA" with a negative update of the covariance matrix and
  96.      * checks for positive definiteness. OPTS.CMA.active = 2 does not check for
  97.      * pos. def. and is numerically faster. Active CMA usually speeds up the
  98.      * adaptation.
  99.      */
  100.     private final boolean isActiveCMA;
  101.     /**
  102.      * Determines how often a new random offspring is generated in case it is
  103.      * not feasible / beyond the defined limits, default is 0.
  104.      */
  105.     private final int checkFeasableCount;
  106.     /**
  107.      * @see Sigma
  108.      */
  109.     private double[] inputSigma;
  110.     /** Number of objective variables/problem dimension */
  111.     private int dimension;
  112.     /**
  113.      * Defines the number of initial iterations, where the covariance matrix
  114.      * remains diagonal and the algorithm has internally linear time complexity.
  115.      * diagonalOnly = 1 means keeping the covariance matrix always diagonal and
  116.      * this setting also exhibits linear space complexity. This can be
  117.      * particularly useful for dimension > 100.
  118.      * @see <a href="http://hal.archives-ouvertes.fr/inria-00287367/en">A Simple Modification in CMA-ES</a>
  119.      */
  120.     private int diagonalOnly;
  121.     /** Number of objective variables/problem dimension */
  122.     private boolean isMinimize = true;
  123.     /** Indicates whether statistic data is collected. */
  124.     private final boolean generateStatistics;

  126.     // termination criteria
  127.     /** Maximal number of iterations allowed. */
  128.     private final int maxIterations;
  129.     /** Limit for fitness value. */
  130.     private final double stopFitness;
  131.     /** Stop if x-changes larger stopTolUpX. */
  132.     private double stopTolUpX;
  133.     /** Stop if x-change smaller stopTolX. */
  134.     private double stopTolX;
  135.     /** Stop if fun-changes smaller stopTolFun. */
  136.     private double stopTolFun;
  137.     /** Stop if back fun-changes smaller stopTolHistFun. */
  138.     private double stopTolHistFun;

  140.     // selection strategy parameters
  141.     /** Number of parents/points for recombination. */
  142.     private int mu; //
  143.     /** log(mu + 0.5), stored for efficiency. */
  144.     private double logMu2;   // NOPMD - using a field here is for performance reasons
  145.     /** Array for weighted recombination. */
  146.     private RealMatrix weights;
  147.     /** Variance-effectiveness of sum w_i x_i. */
  148.     private double mueff; //

  150.     // dynamic strategy parameters and constants
  151.     /** Overall standard deviation - search volume. */
  152.     private double sigma;
  153.     /** Cumulation constant. */
  154.     private double cc;
  155.     /** Cumulation constant for step-size. */
  156.     private double cs;
  157.     /** Damping for step-size. */
  158.     private double damps;
  159.     /** Learning rate for rank-one update. */
  160.     private double ccov1;
  161.     /** Learning rate for rank-mu update' */
  162.     private double ccovmu;
  163.     /** Expectation of ||N(0,I)|| == norm(randn(N,1)). */
  164.     private double chiN;
  165.     /** Learning rate for rank-one update - diagonalOnly */
  166.     private double ccov1Sep;
  167.     /** Learning rate for rank-mu update - diagonalOnly */
  168.     private double ccovmuSep;

  170.     // CMA internal values - updated each generation
  171.     /** Objective variables. */
  172.     private RealMatrix xmean;
  173.     /** Evolution path. */
  174.     private RealMatrix pc;
  175.     /** Evolution path for sigma. */
  176.     private RealMatrix ps;
  177.     /** Norm of ps, stored for efficiency. */
  178.     private double normps;
  179.     /** Coordinate system. */
  180.     private RealMatrix B;
  181.     /** Scaling. */
  182.     private RealMatrix D; // NOPMD
  183.     /** B*D, stored for efficiency. */
  184.     private RealMatrix BD;
  185.     /** Diagonal of sqrt(D), stored for efficiency. */
  186.     private RealMatrix diagD;
  187.     /** Covariance matrix. */
  188.     private RealMatrix C;
  189.     /** Diagonal of C, used for diagonalOnly. */
  190.     private RealMatrix diagC;
  191.     /** Number of iterations already performed. */
  192.     private int iterations;

  194.     /** History queue of best values. */
  195.     private double[] fitnessHistory;

  197.     /** Random generator. */
  198.     private final RandomGenerator random;

  200.     /** History of sigma values. */
  201.     private final List<Double> statisticsSigmaHistory;
  202.     /** History of mean matrix. */
  203.     private final List<RealMatrix> statisticsMeanHistory;
  204.     /** History of fitness values. */
  205.     private final List<Double> statisticsFitnessHistory;
  206.     /** History of D matrix. */
  207.     private final List<RealMatrix> statisticsDHistory;

  209.     /** Simple constructor.
  210.      * @param maxIterations Maximal number of iterations.
  211.      * @param stopFitness Whether to stop if objective function value is smaller than
  212.      * {@code stopFitness}.
  213.      * @param isActiveCMA Chooses the covariance matrix update method.
  214.      * @param diagonalOnly Number of initial iterations, where the covariance matrix
  215.      * remains diagonal.
  216.      * @param checkFeasableCount Determines how often new random objective variables are
  217.      * generated in case they are out of bounds.
  218.      * @param random Random generator.
  219.      * @param generateStatistics Whether statistic data is collected.
  220.      * @param checker Convergence checker.
  221.      *
  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.         this.statisticsSigmaHistory = new ArrayList<>();
  240.         this.statisticsMeanHistory = new ArrayList<>();
  241.         this.statisticsFitnessHistory = new ArrayList<>();
  242.         this.statisticsDHistory = new ArrayList<>();
  243.     }

  245.     /** Get history of sigma values.
  246.      * @return History of sigma values
  247.      */
  248.     public List<Double> getStatisticsSigmaHistory() {
  249.         return statisticsSigmaHistory;
  250.     }

  252.     /** Get history of mean matrix.
  253.      * @return History of mean matrix
  254.      */
  255.     public List<RealMatrix> getStatisticsMeanHistory() {
  256.         return statisticsMeanHistory;
  257.     }

  259.     /** Get history of fitness values.
  260.      * @return History of fitness values
  261.      */
  262.     public List<Double> getStatisticsFitnessHistory() {
  263.         return statisticsFitnessHistory;
  264.     }

  266.     /** Get history of D matrix.
  267.      * @return History of D matrix
  268.      */
  269.     public List<RealMatrix> getStatisticsDHistory() {
  270.         return statisticsDHistory;
  271.     }

  273.     /**
  274.      * Input sigma values.
  275.      * They define the initial coordinate-wise standard deviations for
  276.      * sampling new search points around the initial guess.
  277.      * It is suggested to set them to the estimated distance from the
  278.      * initial to the desired optimum.
  279.      * Small values induce the search to be more local (and very small
  280.      * values are more likely to find a local optimum close to the initial
  281.      * guess).
  282.      * Too small values might however lead to early termination.
  283.      */
  284.     public static class Sigma implements OptimizationData {
  285.         /** Sigma values. */
  286.         private final double[] s;

  288.         /** Simple constructor.
  289.          * @param s Sigma values.
  290.          * @throws MathIllegalArgumentException if any of the array entries is smaller
  291.          * than zero.
  292.          */
  293.         public Sigma(double[] s)
  294.             throws MathIllegalArgumentException {
  295.             for (double v : s) {
  296.                 if (v < 0) {
  297.                     throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL, v, 0);
  298.                 }
  299.             }

  301.             this.s = s.clone();
  302.         }

  304.         /** Get sigma values.
  305.          * @return the sigma values
  306.          */
  307.         public double[] getSigma() {
  308.             return s.clone();
  309.         }
  310.     }

  312.     /**
  313.      * Population size.
  314.      * The number of offspring is the primary strategy parameter.
  315.      * In the absence of better clues, a good default could be an
  316.      * integer close to {@code 4 + 3 ln(n)}, where {@code n} is the
  317.      * number of optimized parameters.
  318.      * Increasing the population size improves global search properties
  319.      * at the expense of speed (which in general decreases at most
  320.      * linearly with increasing population size).
  321.      */
  322.     public static class PopulationSize implements OptimizationData {
  323.         /** Population size. */
  324.         private final int lambda;

  326.         /** Simple constructor.
  327.          * @param size Population size.
  328.          * @throws MathIllegalArgumentException if {@code size <= 0}.
  329.          */
  330.         public PopulationSize(int size)
  331.             throws MathIllegalArgumentException {
  332.             if (size <= 0) {
  333.                 throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL_BOUND_EXCLUDED,
  334.                                                        size, 0);
  335.             }
  336.             lambda = size;
  337.         }

  339.         /** Get population size.
  340.          * @return the population size
  341.          */
  342.         public int getPopulationSize() {
  343.             return lambda;
  344.         }
  345.     }

  347.     /**
  348.      * {@inheritDoc}
  349.      *
  350.      * @param optData Optimization data. In addition to those documented in
  351.      * {@link MultivariateOptimizer#parseOptimizationData(OptimizationData[])
  352.      * MultivariateOptimizer}, this method will register the following data:
  353.      * <ul>
  354.      *  <li>{@link Sigma}</li>
  355.      *  <li>{@link PopulationSize}</li>
  356.      * </ul>
  357.      * @return {@inheritDoc}
  358.      * @throws MathIllegalStateException if the maximal number of
  359.      * evaluations is exceeded.
  360.      * @throws MathIllegalArgumentException if the initial guess, target, and weight
  361.      * arguments have inconsistent dimensions.
  362.      */
  363.     @Override
  364.     public PointValuePair optimize(OptimizationData... optData)
  365.         throws MathIllegalArgumentException, MathIllegalStateException {
  366.         // Set up base class and perform computation.
  367.         return super.optimize(optData);
  368.     }

  370.     /** {@inheritDoc} */
  371.     @Override
  372.     protected PointValuePair doOptimize() {
  373.          // -------------------- Initialization --------------------------------
  374.         isMinimize = getGoalType().equals(GoalType.MINIMIZE);
  375.         final FitnessFunction fitfun = new FitnessFunction();
  376.         final double[] guess = getStartPoint();
  377.         // number of objective variables/problem dimension
  378.         dimension = guess.length;
  379.         initializeCMA(guess);
  380.         ValuePenaltyPair valuePenalty = fitfun.value(guess);
  381.         double bestValue = valuePenalty.value+valuePenalty.penalty;
  382.         push(fitnessHistory, bestValue);
  383.         PointValuePair optimum
  384.             = new PointValuePair(getStartPoint(),
  385.                                  isMinimize ? bestValue : -bestValue);
  386.         PointValuePair lastResult = null;

  388.         // -------------------- Generation Loop --------------------------------

  390.         generationLoop:
  391.         for (iterations = 1; iterations <= maxIterations; iterations++) {
  392.             incrementIterationCount();

  394.             // Generate and evaluate lambda offspring
  395.             final RealMatrix arz = randn1(dimension, lambda);
  396.             final RealMatrix arx = zeros(dimension, lambda);
  397.             final double[] fitness = new double[lambda];
  398.             final ValuePenaltyPair[] valuePenaltyPairs = new ValuePenaltyPair[lambda];
  399.             // generate random offspring
  400.             for (int k = 0; k < lambda; k++) {
  401.                 RealMatrix arxk = null;
  402.                 for (int i = 0; i < checkFeasableCount + 1; i++) {
  403.                     if (diagonalOnly <= 0) {
  404.                         arxk = xmean.add(BD.multiply(arz.getColumnMatrix(k))
  405.                                          .scalarMultiply(sigma)); // m + sig * Normal(0,C)
  406.                     } else {
  407.                         arxk = xmean.add(times(diagD,arz.getColumnMatrix(k))
  408.                                          .scalarMultiply(sigma));
  409.                     }
  410.                     if (i >= checkFeasableCount ||
  411.                         fitfun.isFeasible(arxk.getColumn(0))) {
  412.                         break;
  413.                     }
  414.                     // regenerate random arguments for row
  415.                     arz.setColumn(k, randn(dimension));
  416.                 }
  417.                 copyColumn(arxk, 0, arx, k);
  418.                 try {
  419.                     valuePenaltyPairs[k] = fitfun.value(arx.getColumn(k)); // compute fitness
  420.                 } catch (MathIllegalStateException e) {
  421.                     break generationLoop;
  422.                 }
  423.             }

  425.             // Compute fitnesses by adding value and penalty after scaling by value range.
  426.             double valueRange = valueRange(valuePenaltyPairs);
  427.             for (int iValue=0;iValue<valuePenaltyPairs.length;iValue++) {
  428.                  fitness[iValue] = valuePenaltyPairs[iValue].value + valuePenaltyPairs[iValue].penalty*valueRange;
  429.             }

  431.             // Sort by fitness and compute weighted mean into xmean
  432.             final int[] arindex = sortedIndices(fitness);
  433.             // Calculate new xmean, this is selection and recombination
  434.             final RealMatrix xold = xmean; // for speed up of Eq. (2) and (3)
  435.             final RealMatrix bestArx = selectColumns(arx, Arrays.copyOf(arindex, mu));
  436.             xmean = bestArx.multiply(weights);
  437.             final RealMatrix bestArz = selectColumns(arz, Arrays.copyOf(arindex, mu));
  438.             final RealMatrix zmean = bestArz.multiply(weights);
  439.             final boolean hsig = updateEvolutionPaths(zmean, xold);
  440.             if (diagonalOnly <= 0) {
  441.                 updateCovariance(hsig, bestArx, arz, arindex, xold);
  442.             } else {
  443.                 updateCovarianceDiagonalOnly(hsig, bestArz);
  444.             }
  445.             // Adapt step size sigma - Eq. (5)
  446.             sigma *= FastMath.exp(FastMath.min(1, (normps/chiN - 1) * cs / damps));
  447.             final double bestFitness = fitness[arindex[0]];
  448.             final double worstFitness = fitness[arindex[arindex.length - 1]];
  449.             if (bestValue > bestFitness) {
  450.                 bestValue = bestFitness;
  451.                 lastResult = optimum;
  452.                 optimum = new PointValuePair(fitfun.repair(bestArx.getColumn(0)),
  453.                                              isMinimize ? bestFitness : -bestFitness);
  454.                 if (getConvergenceChecker() != null && lastResult != null &&
  455.                     getConvergenceChecker().converged(iterations, optimum, lastResult)) {
  456.                     break generationLoop;
  457.                 }
  458.             }
  459.             // handle termination criteria
  460.             // Break, if fitness is good enough
  461.             if (stopFitness != 0 && bestFitness < (isMinimize ? stopFitness : -stopFitness)) {
  462.                 break generationLoop;
  463.             }
  464.             final double[] sqrtDiagC = sqrt(diagC).getColumn(0);
  465.             final double[] pcCol = pc.getColumn(0);
  466.             for (int i = 0; i < dimension; i++) {
  467.                 if (sigma * FastMath.max(FastMath.abs(pcCol[i]), sqrtDiagC[i]) > stopTolX) {
  468.                     break;
  469.                 }
  470.                 if (i >= dimension - 1) {
  471.                     break generationLoop;
  472.                 }
  473.             }
  474.             for (int i = 0; i < dimension; i++) {
  475.                 if (sigma * sqrtDiagC[i] > stopTolUpX) {
  476.                     break generationLoop;
  477.                 }
  478.             }
  479.             final double historyBest = min(fitnessHistory);
  480.             final double historyWorst = max(fitnessHistory);
  481.             if (iterations > 2 &&
  482.                 FastMath.max(historyWorst, worstFitness) -
  483.                 FastMath.min(historyBest, bestFitness) < stopTolFun) {
  484.                 break generationLoop;
  485.             }
  486.             if (iterations > fitnessHistory.length &&
  487.                 historyWorst - historyBest < stopTolHistFun) {
  488.                 break generationLoop;
  489.             }
  490.             // condition number of the covariance matrix exceeds 1e14
  491.             if (max(diagD) / min(diagD) > 1e7) {
  492.                 break generationLoop;
  493.             }
  494.             // user defined termination
  495.             if (getConvergenceChecker() != null) {
  496.                 final PointValuePair current
  497.                     = new PointValuePair(bestArx.getColumn(0),
  498.                                          isMinimize ? bestFitness : -bestFitness);
  499.                 if (lastResult != null &&
  500.                     getConvergenceChecker().converged(iterations, current, lastResult)) {
  501.                     break generationLoop;
  502.                     }
  503.                 lastResult = current;
  504.             }
  505.             // Adjust step size in case of equal function values (flat fitness)
  506.             if (bestValue == fitness[arindex[(int)(0.1+lambda/4.)]]) {
  507.                 sigma *= FastMath.exp(0.2 + cs / damps);
  508.             }
  509.             if (iterations > 2 && FastMath.max(historyWorst, bestFitness) -
  510.                 FastMath.min(historyBest, bestFitness) == 0) {
  511.                 sigma *= FastMath.exp(0.2 + cs / damps);
  512.             }
  513.             // store best in history
  514.             push(fitnessHistory,bestFitness);
  515.             if (generateStatistics) {
  516.                 statisticsSigmaHistory.add(sigma);
  517.                 statisticsFitnessHistory.add(bestFitness);
  518.                 statisticsMeanHistory.add(xmean.transpose());
  519.                 statisticsDHistory.add(diagD.transpose().scalarMultiply(1E5));
  520.             }
  521.         }
  522.         return optimum;
  523.     }

  525.     /**
  526.      * Scans the list of (required and optional) optimization data that
  527.      * characterize the problem.
  528.      *
  529.      * @param optData Optimization data. The following data will be looked for:
  530.      * <ul>
  531.      *  <li>{@link Sigma}</li>
  532.      *  <li>{@link PopulationSize}</li>
  533.      * </ul>
  534.      */
  535.     @Override
  536.     protected void parseOptimizationData(OptimizationData... optData) {
  537.         // Allow base class to register its own data.
  538.         super.parseOptimizationData(optData);

  540.         // The existing values (as set by the previous call) are reused if
  541.         // not provided in the argument list.
  542.         for (OptimizationData data : optData) {
  543.             if (data instanceof Sigma) {
  544.                 inputSigma = ((Sigma) data).getSigma();
  545.                 continue;
  546.             }
  547.             if (data instanceof PopulationSize) {
  548.                 lambda = ((PopulationSize) data).getPopulationSize();
  549.                 continue;
  550.             }
  551.         }

  553.         checkParameters();
  554.     }

  556.     /**
  557.      * Checks dimensions and values of boundaries and inputSigma if defined.
  558.      */
  559.     private void checkParameters() {
  560.         if (inputSigma != null) {
  561.             final double[] init = getStartPoint();

  563.             if (inputSigma.length != init.length) {
  564.                 throw new MathIllegalArgumentException(LocalizedCoreFormats.DIMENSIONS_MISMATCH,
  565.                                                        inputSigma.length, init.length);
  566.             }

  568.             final double[] lB = getLowerBound();
  569.             final double[] uB = getUpperBound();

  571.             for (int i = 0; i < init.length; i++) {
  572.                 if (inputSigma[i] > uB[i] - lB[i]) {
  573.                     throw new MathIllegalArgumentException(LocalizedCoreFormats.OUT_OF_RANGE_SIMPLE,
  574.                                                            inputSigma[i], 0, uB[i] - lB[i]);
  575.                 }
  576.             }
  577.         }
  578.     }

  580.     /**
  581.      * Initialization of the dynamic search parameters
  582.      *
  583.      * @param guess Initial guess for the arguments of the fitness function.
  584.      */
  585.     private void initializeCMA(double[] guess) {
  586.         if (lambda <= 0) {
  587.             throw new MathIllegalArgumentException(LocalizedCoreFormats.NUMBER_TOO_SMALL_BOUND_EXCLUDED,
  588.                                                    lambda, 0);
  589.         }
  590.         // initialize sigma
  591.         final double[][] sigmaArray = new double[guess.length][1];
  592.         for (int i = 0; i < guess.length; i++) {
  593.             sigmaArray[i][0] = inputSigma[i];
  594.         }
  595.         final RealMatrix insigma = new Array2DRowRealMatrix(sigmaArray, false);
  596.         sigma = max(insigma); // overall standard deviation

  598.         // initialize termination criteria
  599.         stopTolUpX = 1e3 * max(insigma);
  600.         stopTolX = 1e-11 * max(insigma);
  601.         stopTolFun = 1e-12;
  602.         stopTolHistFun = 1e-13;

  604.         // initialize selection strategy parameters
  605.         mu = lambda / 2; // number of parents/points for recombination
  606.         logMu2 = FastMath.log(mu + 0.5);
  607.         weights = log(sequence(1, mu, 1)).scalarMultiply(-1).scalarAdd(logMu2);
  608.         double sumw = 0;
  609.         double sumwq = 0;
  610.         for (int i = 0; i < mu; i++) {
  611.             double w = weights.getEntry(i, 0);
  612.             sumw += w;
  613.             sumwq += w * w;
  614.         }
  615.         weights = weights.scalarMultiply(1 / sumw);
  616.         mueff = sumw * sumw / sumwq; // variance-effectiveness of sum w_i x_i

  618.         // initialize dynamic strategy parameters and constants
  619.         cc = (4 + mueff / dimension) /
  620.                 (dimension + 4 + 2 * mueff / dimension);
  621.         cs = (mueff + 2) / (dimension + mueff + 3.);
  622.         damps = (1 + 2 * FastMath.max(0, FastMath.sqrt((mueff - 1) /
  623.                                                        (dimension + 1)) - 1)) *
  624.             FastMath.max(0.3,
  625.                          1 - dimension / (1e-6 + maxIterations)) + cs; // minor increment
  626.         ccov1 = 2 / ((dimension + 1.3) * (dimension + 1.3) + mueff);
  627.         ccovmu = FastMath.min(1 - ccov1, 2 * (mueff - 2 + 1 / mueff) /
  628.                               ((dimension + 2) * (dimension + 2) + mueff));
  629.         ccov1Sep = FastMath.min(1, ccov1 * (dimension + 1.5) / 3);
  630.         ccovmuSep = FastMath.min(1 - ccov1, ccovmu * (dimension + 1.5) / 3);
  631.         chiN = FastMath.sqrt(dimension) *
  632.                 (1 - 1 / ((double) 4 * dimension) + 1 / ((double) 21 * dimension * dimension));
  633.         // intialize CMA internal values - updated each generation
  634.         xmean = MatrixUtils.createColumnRealMatrix(guess); // objective variables
  635.         diagD = insigma.scalarMultiply(1 / sigma);
  636.         diagC = square(diagD);
  637.         pc = zeros(dimension, 1); // evolution paths for C and sigma
  638.         ps = zeros(dimension, 1); // B defines the coordinate system
  639.         normps = ps.getFrobeniusNorm();

  641.         B = eye(dimension, dimension);
  642.         D = ones(dimension, 1); // diagonal D defines the scaling
  643.         BD = times(B, repmat(diagD.transpose(), dimension, 1));
  644.         C = B.multiply(diag(square(D)).multiply(B.transpose())); // covariance
  645.         final int historySize = 10 + (int) (3 * 10 * dimension / (double) lambda);
  646.         fitnessHistory = new double[historySize]; // history of fitness values
  647.         for (int i = 0; i < historySize; i++) {
  648.             fitnessHistory[i] = Double.MAX_VALUE;
  649.         }
  650.     }

  652.     /**
  653.      * Update of the evolution paths ps and pc.
  654.      *
  655.      * @param zmean Weighted row matrix of the gaussian random numbers generating
  656.      * the current offspring.
  657.      * @param xold xmean matrix of the previous generation.
  658.      * @return hsig flag indicating a small correction.
  659.      */
  660.     private boolean updateEvolutionPaths(RealMatrix zmean, RealMatrix xold) {
  661.         ps = ps.scalarMultiply(1 - cs).add(
  662.                 B.multiply(zmean).scalarMultiply(
  663.                         FastMath.sqrt(cs * (2 - cs) * mueff)));
  664.         normps = ps.getFrobeniusNorm();
  665.         final boolean hsig = normps /
  666.             FastMath.sqrt(1 - FastMath.pow(1 - cs, 2 * iterations)) /
  667.             chiN < 1.4 + 2 / ((double) dimension + 1);
  668.         pc = pc.scalarMultiply(1 - cc);
  669.         if (hsig) {
  670.             pc = pc.add(xmean.subtract(xold).scalarMultiply(FastMath.sqrt(cc * (2 - cc) * mueff) / sigma));
  671.         }
  672.         return hsig;
  673.     }

  675.     /**
  676.      * Update of the covariance matrix C for diagonalOnly > 0
  677.      *
  678.      * @param hsig Flag indicating a small correction.
  679.      * @param bestArz Fitness-sorted matrix of the gaussian random values of the
  680.      * current offspring.
  681.      */
  682.     private void updateCovarianceDiagonalOnly(boolean hsig,
  683.                                               final RealMatrix bestArz) {
  684.         // minor correction if hsig==false
  685.         double oldFac = hsig ? 0 : ccov1Sep * cc * (2 - cc);
  686.         oldFac += 1 - ccov1Sep - ccovmuSep;
  687.         diagC = diagC.scalarMultiply(oldFac) // regard old matrix
  688.             .add(square(pc).scalarMultiply(ccov1Sep)) // plus rank one update
  689.             .add(times(diagC, square(bestArz).multiply(weights)) // plus rank mu update
  690.                  .scalarMultiply(ccovmuSep));
  691.         diagD = sqrt(diagC); // replaces eig(C)
  692.         if (diagonalOnly > 1 &&
  693.             iterations > diagonalOnly) {
  694.             // full covariance matrix from now on
  695.             diagonalOnly = 0;
  696.             B = eye(dimension, dimension);
  697.             BD = diag(diagD);
  698.             C = diag(diagC);
  699.         }
  700.     }

  702.     /**
  703.      * Update of the covariance matrix C.
  704.      *
  705.      * @param hsig Flag indicating a small correction.
  706.      * @param bestArx Fitness-sorted matrix of the argument vectors producing the
  707.      * current offspring.
  708.      * @param arz Unsorted matrix containing the gaussian random values of the
  709.      * current offspring.
  710.      * @param arindex Indices indicating the fitness-order of the current offspring.
  711.      * @param xold xmean matrix of the previous generation.
  712.      */
  713.     private void updateCovariance(boolean hsig, final RealMatrix bestArx,
  714.                                   final RealMatrix arz, final int[] arindex,
  715.                                   final RealMatrix xold) {
  716.         double negccov = 0;
  717.         if (ccov1 + ccovmu > 0) {
  718.             final RealMatrix arpos = bestArx.subtract(repmat(xold, 1, mu))
  719.                 .scalarMultiply(1 / sigma); // mu difference vectors
  720.             final RealMatrix roneu = pc.multiplyTransposed(pc)
  721.                 .scalarMultiply(ccov1); // rank one update
  722.             // minor correction if hsig==false
  723.             double oldFac = hsig ? 0 : ccov1 * cc * (2 - cc);
  724.             oldFac += 1 - ccov1 - ccovmu;
  725.             if (isActiveCMA) {
  726.                 // Adapt covariance matrix C active CMA
  727.                 negccov = (1 - ccovmu) * 0.25 * mueff /
  728.                     (FastMath.pow(dimension + 2, 1.5) + 2 * mueff);
  729.                 // keep at least 0.66 in all directions, small popsize are most
  730.                 // critical
  731.                 final double negminresidualvariance = 0.66;
  732.                 // where to make up for the variance loss
  733.                 final double negalphaold = 0.5;
  734.                 // prepare vectors, compute negative updating matrix Cneg
  735.                 final int[] arReverseIndex = reverse(arindex);
  736.                 RealMatrix arzneg = selectColumns(arz, Arrays.copyOf(arReverseIndex, mu));
  737.                 RealMatrix arnorms = sqrt(sumRows(square(arzneg)));
  738.                 final int[] idxnorms = sortedIndices(arnorms.getRow(0));
  739.                 final RealMatrix arnormsSorted = selectColumns(arnorms, idxnorms);
  740.                 final int[] idxReverse = reverse(idxnorms);
  741.                 final RealMatrix arnormsReverse = selectColumns(arnorms, idxReverse);
  742.                 arnorms = divide(arnormsReverse, arnormsSorted);
  743.                 final int[] idxInv = inverse(idxnorms);
  744.                 final RealMatrix arnormsInv = selectColumns(arnorms, idxInv);
  745.                 // check and set learning rate negccov
  746.                 final double negcovMax = (1 - negminresidualvariance) /
  747.                     square(arnormsInv).multiply(weights).getEntry(0, 0);
  748.                 if (negccov > negcovMax) {
  749.                     negccov = negcovMax;
  750.                 }
  751.                 arzneg = times(arzneg, repmat(arnormsInv, dimension, 1));
  752.                 final RealMatrix artmp = BD.multiply(arzneg);
  753.                 final RealMatrix Cneg = artmp.multiply(diag(weights)).multiply(artmp.transpose());
  754.                 oldFac += negalphaold * negccov;
  755.                 C = C.scalarMultiply(oldFac)
  756.                     .add(roneu) // regard old matrix
  757.                     .add(arpos.scalarMultiply( // plus rank one update
  758.                                               ccovmu + (1 - negalphaold) * negccov) // plus rank mu update
  759.                          .multiply(times(repmat(weights, 1, dimension),
  760.                                          arpos.transpose())))
  761.                     .subtract(Cneg.scalarMultiply(negccov));
  762.             } else {
  763.                 // Adapt covariance matrix C - nonactive
  764.                 C = C.scalarMultiply(oldFac) // regard old matrix
  765.                     .add(roneu) // plus rank one update
  766.                     .add(arpos.scalarMultiply(ccovmu) // plus rank mu update
  767.                          .multiply(times(repmat(weights, 1, dimension),
  768.                                          arpos.transpose())));
  769.             }
  770.         }
  771.         updateBD(negccov);
  772.     }

  774.     /**
  775.      * Update B and D from C.
  776.      *
  777.      * @param negccov Negative covariance factor.
  778.      */
  779.     private void updateBD(double negccov) {
  780.         if (ccov1 + ccovmu + negccov > 0 &&
  781.             (iterations % 1. / (ccov1 + ccovmu + negccov) / dimension / 10.) < 1) {
  782.             // to achieve O(N^2)
  783.             C = triu(C, 0).add(triu(C, 1).transpose());
  784.             // enforce symmetry to prevent complex numbers
  785.             final EigenDecompositionSymmetric eig = new EigenDecompositionSymmetric(C);
  786.             B = eig.getV(); // eigen decomposition, B==normalized eigenvectors
  787.             D = eig.getD();
  788.             diagD = diag(D);
  789.             if (min(diagD) <= 0) {
  790.                 for (int i = 0; i < dimension; i++) {
  791.                     if (diagD.getEntry(i, 0) < 0) {
  792.                         diagD.setEntry(i, 0, 0);
  793.                     }
  794.                 }
  795.                 final double tfac = max(diagD) / 1e14;
  796.                 C = C.add(eye(dimension, dimension).scalarMultiply(tfac));
  797.                 diagD = diagD.add(ones(dimension, 1).scalarMultiply(tfac));
  798.             }
  799.             if (max(diagD) > 1e14 * min(diagD)) {
  800.                 final double tfac = max(diagD) / 1e14 - min(diagD);
  801.                 C = C.add(eye(dimension, dimension).scalarMultiply(tfac));
  802.                 diagD = diagD.add(ones(dimension, 1).scalarMultiply(tfac));
  803.             }
  804.             diagC = diag(C);
  805.             diagD = sqrt(diagD); // D contains standard deviations now
  806.             BD = times(B, repmat(diagD.transpose(), dimension, 1)); // O(n^2)
  807.         }
  808.     }

  810.     /**
  811.      * Pushes the current best fitness value in a history queue.
  812.      *
  813.      * @param vals History queue.
  814.      * @param val Current best fitness value.
  815.      */
  816.     private static void push(double[] vals, double val) {
  817.         for (int i = vals.length-1; i > 0; i--) {
  818.             vals[i] = vals[i-1];
  819.         }
  820.         vals[0] = val;
  821.     }

  823.     /**
  824.      * Sorts fitness values.
  825.      *
  826.      * @param doubles Array of values to be sorted.
  827.      * @return a sorted array of indices pointing into doubles.
  828.      */
  829.     private int[] sortedIndices(final double[] doubles) {
  830.         final DoubleIndex[] dis = new DoubleIndex[doubles.length];
  831.         for (int i = 0; i < doubles.length; i++) {
  832.             dis[i] = new DoubleIndex(doubles[i], i);
  833.         }
  834.         Arrays.sort(dis);
  835.         final int[] indices = new int[doubles.length];
  836.         for (int i = 0; i < doubles.length; i++) {
  837.             indices[i] = dis[i].index;
  838.         }
  839.         return indices;
  840.     }
  841.    /**
  842.      * Get range of values.
  843.      *
  844.      * @param vpPairs Array of valuePenaltyPairs to get range from.
  845.      * @return a double equal to maximum value minus minimum value.
  846.      */
  847.     private double valueRange(final ValuePenaltyPair[] vpPairs) {
  848.         double max = Double.NEGATIVE_INFINITY;
  849.         double min = Double.MAX_VALUE;
  850.         for (ValuePenaltyPair vpPair:vpPairs) {
  851.             if (vpPair.value > max) {
  852.                 max = vpPair.value;
  853.             }
  854.             if (vpPair.value < min) {
  855.                 min = vpPair.value;
  856.             }
  857.         }
  858.         return max-min;
  859.     }

  861.     /**
  862.      * Used to sort fitness values. Sorting is always in lower value first
  863.      * order.
  864.      */
  865.     private static class DoubleIndex implements Comparable<DoubleIndex> {
  866.         /** Value to compare. */
  867.         private final double value;
  868.         /** Index into sorted array. */
  869.         private final int index;

  871.         /**
  872.          * @param value Value to compare.
  873.          * @param index Index into sorted array.
  874.          */
  875.         DoubleIndex(double value, int index) {
  876.             this.value = value;
  877.             this.index = index;
  878.         }

  880.         /** {@inheritDoc} */
  881.         @Override
  882.         public int compareTo(DoubleIndex o) {
  883.             return Double.compare(value, o.value);
  884.         }

  886.         /** {@inheritDoc} */
  887.         @Override
  888.         public boolean equals(Object other) {

  890.             if (this == other) {
  891.                 return true;
  892.             }

  894.             if (other instanceof DoubleIndex) {
  895.                 return Double.compare(value, ((DoubleIndex) other).value) == 0;
  896.             }

  898.             return false;
  899.         }

  901.         /** {@inheritDoc} */
  902.         @Override
  903.         public int hashCode() {
  904.             long bits = Double.doubleToLongBits(value);
  905.             return (int) ((1438542 ^ (bits >>> 32) ^ bits));
  906.         }
  907.     }
  908.     /**
  909.      * Stores the value and penalty (for repair of out of bounds point).
  910.      */
  911.     private static class ValuePenaltyPair {
  912.         /** Objective function value. */
  913.         private double value;
  914.         /** Penalty value for repair of out out of bounds points. */
  915.         private double penalty;

  917.         /**
  918.          * @param value Function value.
  919.          * @param penalty Out-of-bounds penalty.
  920.         */
  921.         ValuePenaltyPair(final double value, final double penalty) {
  922.             this.value   = value;
  923.             this.penalty = penalty;
  924.         }
  925.     }


  928.     /**
  929.      * Normalizes fitness values to the range [0,1]. Adds a penalty to the
  930.      * fitness value if out of range.
  931.      */
  932.     private class FitnessFunction {
  933.         /**
  934.          * Flag indicating whether the objective variables are forced into their
  935.          * bounds if defined
  936.          */
  937.         private final boolean isRepairMode;

  939.         /** Simple constructor.
  940.          */
  941.         FitnessFunction() {
  942.             isRepairMode = true;
  943.         }

  945.         /**
  946.          * @param point Normalized objective variables.
  947.          * @return the objective value + penalty for violated bounds.
  948.          */
  949.         public ValuePenaltyPair value(final double[] point) {
  950.             double value;
  951.             double penalty=0.0;
  952.             if (isRepairMode) {
  953.                 double[] repaired = repair(point);
  954.                 value = CMAESOptimizer.this.computeObjectiveValue(repaired);
  955.                 penalty =  penalty(point, repaired);
  956.             } else {
  957.                 value = CMAESOptimizer.this.computeObjectiveValue(point);
  958.             }
  959.             value = isMinimize ? value : -value;
  960.             penalty = isMinimize ? penalty : -penalty;
  961.             return new ValuePenaltyPair(value,penalty);
  962.         }

  964.         /**
  965.          * @param x Normalized objective variables.
  966.          * @return {@code true} if in bounds.
  967.          */
  968.         public boolean isFeasible(final double[] x) {
  969.             final double[] lB = CMAESOptimizer.this.getLowerBound();
  970.             final double[] uB = CMAESOptimizer.this.getUpperBound();

  972.             for (int i = 0; i < x.length; i++) {
  973.                 if (x[i] < lB[i]) {
  974.                     return false;
  975.                 }
  976.                 if (x[i] > uB[i]) {
  977.                     return false;
  978.                 }
  979.             }
  980.             return true;
  981.         }

  983.         /**
  984.          * @param x Normalized objective variables.
  985.          * @return the repaired (i.e. all in bounds) objective variables.
  986.          */
  987.         private double[] repair(final double[] x) {
  988.             final double[] lB = CMAESOptimizer.this.getLowerBound();
  989.             final double[] uB = CMAESOptimizer.this.getUpperBound();

  991.             final double[] repaired = new double[x.length];
  992.             for (int i = 0; i < x.length; i++) {
  993.                 if (x[i] < lB[i]) {
  994.                     repaired[i] = lB[i];
  995.                 } else if (x[i] > uB[i]) {
  996.                     repaired[i] = uB[i];
  997.                 } else {
  998.                     repaired[i] = x[i];
  999.                 }
  1000.             }
  1001.             return repaired;
  1002.         }

  1004.         /**
  1005.          * @param x Normalized objective variables.
  1006.          * @param repaired Repaired objective variables.
  1007.          * @return Penalty value according to the violation of the bounds.
  1008.          */
  1009.         private double penalty(final double[] x, final double[] repaired) {
  1010.             double penalty = 0;
  1011.             for (int i = 0; i < x.length; i++) {
  1012.                 double diff = FastMath.abs(x[i] - repaired[i]);
  1013.                 penalty += diff;
  1014.             }
  1015.             return isMinimize ? penalty : -penalty;
  1016.         }
  1017.     }

  1019.     // -----Matrix utility functions similar to the Matlab build in functions------

  1021.     /**
  1022.      * @param m Input matrix
  1023.      * @return Matrix representing the element-wise logarithm of m.
  1024.      */
  1025.     private static RealMatrix log(final RealMatrix m) {
  1026.         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
  1027.         for (int r = 0; r < m.getRowDimension(); r++) {
  1028.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1029.                 d[r][c] = FastMath.log(m.getEntry(r, c));
  1030.             }
  1031.         }
  1032.         return new Array2DRowRealMatrix(d, false);
  1033.     }

  1035.     /**
  1036.      * @param m Input matrix.
  1037.      * @return Matrix representing the element-wise square root of m.
  1038.      */
  1039.     private static RealMatrix sqrt(final RealMatrix m) {
  1040.         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
  1041.         for (int r = 0; r < m.getRowDimension(); r++) {
  1042.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1043.                 d[r][c] = FastMath.sqrt(m.getEntry(r, c));
  1044.             }
  1045.         }
  1046.         return new Array2DRowRealMatrix(d, false);
  1047.     }

  1049.     /**
  1050.      * @param m Input matrix.
  1051.      * @return Matrix representing the element-wise square of m.
  1052.      */
  1053.     private static RealMatrix square(final RealMatrix m) {
  1054.         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
  1055.         for (int r = 0; r < m.getRowDimension(); r++) {
  1056.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1057.                 double e = m.getEntry(r, c);
  1058.                 d[r][c] = e * e;
  1059.             }
  1060.         }
  1061.         return new Array2DRowRealMatrix(d, false);
  1062.     }

  1064.     /**
  1065.      * @param m Input matrix 1.
  1066.      * @param n Input matrix 2.
  1067.      * @return the matrix where the elements of m and n are element-wise multiplied.
  1068.      */
  1069.     private static RealMatrix times(final RealMatrix m, final RealMatrix n) {
  1070.         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
  1071.         for (int r = 0; r < m.getRowDimension(); r++) {
  1072.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1073.                 d[r][c] = m.getEntry(r, c) * n.getEntry(r, c);
  1074.             }
  1075.         }
  1076.         return new Array2DRowRealMatrix(d, false);
  1077.     }

  1079.     /**
  1080.      * @param m Input matrix 1.
  1081.      * @param n Input matrix 2.
  1082.      * @return Matrix where the elements of m and n are element-wise divided.
  1083.      */
  1084.     private static RealMatrix divide(final RealMatrix m, final RealMatrix n) {
  1085.         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
  1086.         for (int r = 0; r < m.getRowDimension(); r++) {
  1087.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1088.                 d[r][c] = m.getEntry(r, c) / n.getEntry(r, c);
  1089.             }
  1090.         }
  1091.         return new Array2DRowRealMatrix(d, false);
  1092.     }

  1094.     /**
  1095.      * @param m Input matrix.
  1096.      * @param cols Columns to select.
  1097.      * @return Matrix representing the selected columns.
  1098.      */
  1099.     private static RealMatrix selectColumns(final RealMatrix m, final int[] cols) {
  1100.         final double[][] d = new double[m.getRowDimension()][cols.length];
  1101.         for (int r = 0; r < m.getRowDimension(); r++) {
  1102.             for (int c = 0; c < cols.length; c++) {
  1103.                 d[r][c] = m.getEntry(r, cols[c]);
  1104.             }
  1105.         }
  1106.         return new Array2DRowRealMatrix(d, false);
  1107.     }

  1109.     /**
  1110.      * @param m Input matrix.
  1111.      * @param k Diagonal position.
  1112.      * @return Upper triangular part of matrix.
  1113.      */
  1114.     private static RealMatrix triu(final RealMatrix m, int k) {
  1115.         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
  1116.         for (int r = 0; r < m.getRowDimension(); r++) {
  1117.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1118.                 d[r][c] = r <= c - k ? m.getEntry(r, c) : 0;
  1119.             }
  1120.         }
  1121.         return new Array2DRowRealMatrix(d, false);
  1122.     }

  1124.     /**
  1125.      * @param m Input matrix.
  1126.      * @return Row matrix representing the sums of the rows.
  1127.      */
  1128.     private static RealMatrix sumRows(final RealMatrix m) {
  1129.         final double[][] d = new double[1][m.getColumnDimension()];
  1130.         for (int c = 0; c < m.getColumnDimension(); c++) {
  1131.             double sum = 0;
  1132.             for (int r = 0; r < m.getRowDimension(); r++) {
  1133.                 sum += m.getEntry(r, c);
  1134.             }
  1135.             d[0][c] = sum;
  1136.         }
  1137.         return new Array2DRowRealMatrix(d, false);
  1138.     }

  1140.     /**
  1141.      * @param m Input matrix.
  1142.      * @return the diagonal n-by-n matrix if m is a column matrix or the column
  1143.      * matrix representing the diagonal if m is a n-by-n matrix.
  1144.      */
  1145.     private static RealMatrix diag(final RealMatrix m) {
  1146.         if (m.getColumnDimension() == 1) {
  1147.             final double[][] d = new double[m.getRowDimension()][m.getRowDimension()];
  1148.             for (int i = 0; i < m.getRowDimension(); i++) {
  1149.                 d[i][i] = m.getEntry(i, 0);
  1150.             }
  1151.             return new Array2DRowRealMatrix(d, false);
  1152.         } else {
  1153.             final double[][] d = new double[m.getRowDimension()][1];
  1154.             for (int i = 0; i < m.getColumnDimension(); i++) {
  1155.                 d[i][0] = m.getEntry(i, i);
  1156.             }
  1157.             return new Array2DRowRealMatrix(d, false);
  1158.         }
  1159.     }

  1161.     /**
  1162.      * Copies a column from m1 to m2.
  1163.      *
  1164.      * @param m1 Source matrix.
  1165.      * @param col1 Source column.
  1166.      * @param m2 Target matrix.
  1167.      * @param col2 Target column.
  1168.      */
  1169.     private static void copyColumn(final RealMatrix m1, int col1,
  1170.                                    RealMatrix m2, int col2) {
  1171.         for (int i = 0; i < m1.getRowDimension(); i++) {
  1172.             m2.setEntry(i, col2, m1.getEntry(i, col1));
  1173.         }
  1174.     }

  1176.     /**
  1177.      * @param n Number of rows.
  1178.      * @param m Number of columns.
  1179.      * @return n-by-m matrix filled with 1.
  1180.      */
  1181.     private static RealMatrix ones(int n, int m) {
  1182.         final double[][] d = new double[n][m];
  1183.         for (int r = 0; r < n; r++) {
  1184.             Arrays.fill(d[r], 1);
  1185.         }
  1186.         return new Array2DRowRealMatrix(d, false);
  1187.     }

  1189.     /**
  1190.      * @param n Number of rows.
  1191.      * @param m Number of columns.
  1192.      * @return n-by-m matrix of 0 values out of diagonal, and 1 values on
  1193.      * the diagonal.
  1194.      */
  1195.     private static RealMatrix eye(int n, int m) {
  1196.         final double[][] d = new double[n][m];
  1197.         for (int r = 0; r < n; r++) {
  1198.             if (r < m) {
  1199.                 d[r][r] = 1;
  1200.             }
  1201.         }
  1202.         return new Array2DRowRealMatrix(d, false);
  1203.     }

  1205.     /**
  1206.      * @param n Number of rows.
  1207.      * @param m Number of columns.
  1208.      * @return n-by-m matrix of zero values.
  1209.      */
  1210.     private static RealMatrix zeros(int n, int m) {
  1211.         return new Array2DRowRealMatrix(n, m);
  1212.     }

  1214.     /**
  1215.      * @param mat Input matrix.
  1216.      * @param n Number of row replicates.
  1217.      * @param m Number of column replicates.
  1218.      * @return a matrix which replicates the input matrix in both directions.
  1219.      */
  1220.     private static RealMatrix repmat(final RealMatrix mat, int n, int m) {
  1221.         final int rd = mat.getRowDimension();
  1222.         final int cd = mat.getColumnDimension();
  1223.         final double[][] d = new double[n * rd][m * cd];
  1224.         for (int r = 0; r < n * rd; r++) {
  1225.             for (int c = 0; c < m * cd; c++) {
  1226.                 d[r][c] = mat.getEntry(r % rd, c % cd);
  1227.             }
  1228.         }
  1229.         return new Array2DRowRealMatrix(d, false);
  1230.     }

  1232.     /**
  1233.      * @param start Start value.
  1234.      * @param end End value.
  1235.      * @param step Step size.
  1236.      * @return a sequence as column matrix.
  1237.      */
  1238.     private static RealMatrix sequence(double start, double end, double step) {
  1239.         final int size = (int) ((end - start) / step + 1);
  1240.         final double[][] d = new double[size][1];
  1241.         double value = start;
  1242.         for (int r = 0; r < size; r++) {
  1243.             d[r][0] = value;
  1244.             value += step;
  1245.         }
  1246.         return new Array2DRowRealMatrix(d, false);
  1247.     }

  1249.     /**
  1250.      * @param m Input matrix.
  1251.      * @return the maximum of the matrix element values.
  1252.      */
  1253.     private static double max(final RealMatrix m) {
  1254.         double max = -Double.MAX_VALUE;
  1255.         for (int r = 0; r < m.getRowDimension(); r++) {
  1256.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1257.                 double e = m.getEntry(r, c);
  1258.                 if (max < e) {
  1259.                     max = e;
  1260.                 }
  1261.             }
  1262.         }
  1263.         return max;
  1264.     }

  1266.     /**
  1267.      * @param m Input matrix.
  1268.      * @return the minimum of the matrix element values.
  1269.      */
  1270.     private static double min(final RealMatrix m) {
  1271.         double min = Double.MAX_VALUE;
  1272.         for (int r = 0; r < m.getRowDimension(); r++) {
  1273.             for (int c = 0; c < m.getColumnDimension(); c++) {
  1274.                 double e = m.getEntry(r, c);
  1275.                 if (min > e) {
  1276.                     min = e;
  1277.                 }
  1278.             }
  1279.         }
  1280.         return min;
  1281.     }

  1283.     /**
  1284.      * @param m Input array.
  1285.      * @return the maximum of the array values.
  1286.      */
  1287.     private static double max(final double[] m) {
  1288.         double max = -Double.MAX_VALUE;
  1289.         for (double v : m) {
  1290.             if (max < v) {
  1291.                 max = v;
  1292.             }
  1293.         }
  1294.         return max;
  1295.     }

  1297.     /**
  1298.      * @param m Input array.
  1299.      * @return the minimum of the array values.
  1300.      */
  1301.     private static double min(final double[] m) {
  1302.         double min = Double.MAX_VALUE;
  1303.         for (double v : m) {
  1304.             if (min > v) {
  1305.                 min = v;
  1306.             }
  1307.         }
  1308.         return min;
  1309.     }

  1311.     /**
  1312.      * @param indices Input index array.
  1313.      * @return the inverse of the mapping defined by indices.
  1314.      */
  1315.     private static int[] inverse(final int[] indices) {
  1316.         final int[] inverse = new int[indices.length];
  1317.         for (int i = 0; i < indices.length; i++) {
  1318.             inverse[indices[i]] = i;
  1319.         }
  1320.         return inverse;
  1321.     }

  1323.     /**
  1324.      * @param indices Input index array.
  1325.      * @return the indices in inverse order (last is first).
  1326.      */
  1327.     private static int[] reverse(final int[] indices) {
  1328.         final int[] reverse = new int[indices.length];
  1329.         for (int i = 0; i < indices.length; i++) {
  1330.             reverse[i] = indices[indices.length - i - 1];
  1331.         }
  1332.         return reverse;
  1333.     }

  1335.     /**
  1336.      * @param size Length of random array.
  1337.      * @return an array of Gaussian random numbers.
  1338.      */
  1339.     private double[] randn(int size) {
  1340.         final double[] randn = new double[size];
  1341.         for (int i = 0; i < size; i++) {
  1342.             randn[i] = random.nextGaussian();
  1343.         }
  1344.         return randn;
  1345.     }

  1347.     /**
  1348.      * @param size Number of rows.
  1349.      * @param popSize Population size.
  1350.      * @return a 2-dimensional matrix of Gaussian random numbers.
  1351.      */
  1352.     private RealMatrix randn1(int size, int popSize) {
  1353.         final double[][] d = new double[size][popSize];
  1354.         for (int r = 0; r < size; r++) {
  1355.             for (int c = 0; c < popSize; c++) {
  1356.                 d[r][c] = random.nextGaussian();
  1357.             }
  1358.         }
  1359.         return new Array2DRowRealMatrix(d, false);
  1360.     }
  1361. }
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