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    * Licensed to the Apache Software Foundation (ASF) under one or more
    * contributor license agreements.  See the NOTICE file distributed with
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    * The ASF licenses this file to You under the Apache License, Version 2.0
    * (the "License"); you may not use this file except in compliance with
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    *
    *      http://www.apache.org/licenses/LICENSE-2.0
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  package org.apache.commons.math4.legacy.optim.nonlinear.scalar.noderiv;
  
  import java.util.ArrayList;
  import java.util.Arrays;
  import java.util.List;
  
  import org.apache.commons.math4.legacy.analysis.MultivariateFunction;
  import org.apache.commons.math4.legacy.exception.DimensionMismatchException;
  import org.apache.commons.math4.legacy.exception.NotStrictlyPositiveException;
  import org.apache.commons.math4.legacy.exception.OutOfRangeException;
  import org.apache.commons.math4.legacy.exception.TooManyEvaluationsException;
  import org.apache.commons.math4.legacy.linear.Array2DRowRealMatrix;
  import org.apache.commons.math4.legacy.linear.EigenDecomposition;
  import org.apache.commons.math4.legacy.linear.MatrixUtils;
  import org.apache.commons.math4.legacy.linear.RealMatrix;
  import org.apache.commons.math4.legacy.optim.ConvergenceChecker;
  import org.apache.commons.math4.legacy.optim.OptimizationData;
  import org.apache.commons.math4.legacy.optim.PointValuePair;
  import org.apache.commons.math4.legacy.optim.nonlinear.scalar.GoalType;
  import org.apache.commons.math4.legacy.optim.nonlinear.scalar.PopulationSize;
  import org.apache.commons.math4.legacy.optim.nonlinear.scalar.Sigma;
  import org.apache.commons.math4.legacy.optim.nonlinear.scalar.MultivariateOptimizer;
  import org.apache.commons.rng.UniformRandomProvider;
  import org.apache.commons.statistics.distribution.ContinuousDistribution;
  import org.apache.commons.statistics.distribution.NormalDistribution;
  import org.apache.commons.math4.core.jdkmath.JdkMath;
  
  /**
   * An implementation of the active Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
   * for non-linear, non-convex, non-smooth, global function minimization.
   * <p>
   * The CMA-Evolution Strategy (CMA-ES) is a reliable stochastic optimization method
   * which should be applied if derivative-based methods, e.g. quasi-Newton BFGS or
   * conjugate gradient, fail due to a rugged search landscape (e.g. noise, local
   * optima, outlier, etc.) of the objective function. Like a
   * quasi-Newton method, the CMA-ES learns and applies a variable metric
   * on the underlying search space. Unlike a quasi-Newton method, the
   * CMA-ES neither estimates nor uses gradients, making it considerably more
   * reliable in terms of finding a good, or even close to optimal, solution.
   * <p>
   * In general, on smooth objective functions the CMA-ES is roughly ten times
   * slower than BFGS (counting objective function evaluations, no gradients provided).
   * For up to <code>N=10</code> variables also the derivative-free simplex
   * direct search method (Nelder and Mead) can be faster, but it is
   * far less reliable than CMA-ES.
   * <p>
   * The CMA-ES is particularly well suited for non-separable
   * and/or badly conditioned problems. To observe the advantage of CMA compared
   * to a conventional evolution strategy, it will usually take about
   * <code>30 N</code> function evaluations. On difficult problems the complete
   * optimization (a single run) is expected to take <em>roughly</em> between
   * <code>30 N</code> and <code>300 N<sup>2</sup></code>
   * function evaluations.
   * <p>
   * This implementation is translated and adapted from the Matlab version
   * of the CMA-ES algorithm as implemented in module {@code cmaes.m} version 3.51.
   * <p>
   * For more information, please refer to the following links:
   * <ul>
   *  <li><a href="http://www.lri.fr/~hansen/cmaes.m">Matlab code</a></li>
   *  <li><a href="http://www.lri.fr/~hansen/cmaesintro.html">Introduction to CMA-ES</a></li>
   *  <li><a href="http://en.wikipedia.org/wiki/CMA-ES">Wikipedia</a></li>
   * </ul>
   *
   * @since 3.0
   */
  public class CMAESOptimizer
      extends MultivariateOptimizer {
      // global search parameters
      /**
       * Population size, offspring number. The primary strategy parameter to play
       * with, which can be increased from its default value. Increasing the
       * population size improves global search properties in exchange to speed.
       * Speed decreases, as a rule, at most linearly with increasing population
       * size. It is advisable to begin with the default small population size.
       */
      private int lambda; // population size
      /**
       * Covariance update mechanism, default is active CMA. isActiveCMA = true
       * turns on "active CMA" with a negative update of the covariance matrix and
       * checks for positive definiteness. OPTS.CMA.active = 2 does not check for
       * pos. def. and is numerically faster. Active CMA usually speeds up the
      * adaptation.
      */
     private final boolean isActiveCMA;
     /**
      * Determines how often a new random offspring is generated in case it is
      * not feasible / beyond the defined limits, default is 0.
      */
     private final int checkFeasableCount;
     /**
      * @see Sigma
      */
     private double[] inputSigma;
     /** Number of objective variables/problem dimension. */
     private int dimension;
     /**
      * Defines the number of initial iterations, where the covariance matrix
      * remains diagonal and the algorithm has internally linear time complexity.
      * diagonalOnly = 1 means keeping the covariance matrix always diagonal and
      * this setting also exhibits linear space complexity. This can be
      * particularly useful for dimension > 100.
      * @see <a href="http://hal.archives-ouvertes.fr/inria-00287367/en">A Simple Modification in CMA-ES</a>
      */
     private int diagonalOnly;
     /** Number of objective variables/problem dimension. */
     private boolean isMinimize = true;
     /** Indicates whether statistic data is collected. */
     private final boolean generateStatistics;
 
     // termination criteria
     /** Maximal number of iterations allowed. */
     private final int maxIterations;
     /** Limit for fitness value. */
     private final double stopFitness;
     /** Stop if x-changes larger stopTolUpX. */
     private double stopTolUpX;
     /** Stop if x-change smaller stopTolX. */
     private double stopTolX;
     /** Stop if fun-changes smaller stopTolFun. */
     private double stopTolFun;
     /** Stop if back fun-changes smaller stopTolHistFun. */
     private double stopTolHistFun;
 
     // selection strategy parameters
     /** Number of parents/points for recombination. */
     private int mu; //
     /** log(mu + 0.5), stored for efficiency. */
     private double logMu2;
     /** Array for weighted recombination. */
     private RealMatrix weights;
     /** Variance-effectiveness of sum w_i x_i. */
     private double mueff; //
 
     // dynamic strategy parameters and constants
     /** Overall standard deviation - search volume. */
     private double sigma;
     /** Cumulation constant. */
     private double cc;
     /** Cumulation constant for step-size. */
     private double cs;
     /** Damping for step-size. */
     private double damps;
     /** Learning rate for rank-one update. */
     private double ccov1;
     /** Learning rate for rank-mu update'. */
     private double ccovmu;
     /** Expectation of ||N(0,I)|| == norm(randn(N,1)). */
     private double chiN;
     /** Learning rate for rank-one update - diagonalOnly. */
     private double ccov1Sep;
     /** Learning rate for rank-mu update - diagonalOnly. */
     private double ccovmuSep;
 
     // CMA internal values - updated each generation
     /** Objective variables. */
     private RealMatrix xmean;
     /** Evolution path. */
     private RealMatrix pc;
     /** Evolution path for sigma. */
     private RealMatrix ps;
     /** Norm of ps, stored for efficiency. */
     private double normps;
     /** Coordinate system. */
     private RealMatrix B;
     /** Scaling. */
     private RealMatrix D;
     /** B*D, stored for efficiency. */
     private RealMatrix BD;
     /** Diagonal of sqrt(D), stored for efficiency. */
     private RealMatrix diagD;
     /** Covariance matrix. */
     private RealMatrix C;
     /** Diagonal of C, used for diagonalOnly. */
     private RealMatrix diagC;
     /** Number of iterations already performed. */
     private int iterations;
 
     /** History queue of best values. */
     private double[] fitnessHistory;
     /** Size of history queue of best values. */
     private int historySize;
 
     /** Gaussian sampler. */
     private final ContinuousDistribution.Sampler random;
 
     /** History of sigma values. */
     private final List<Double> statisticsSigmaHistory = new ArrayList<>();
     /** History of mean matrix. */
     private final List<RealMatrix> statisticsMeanHistory = new ArrayList<>();
     /** History of fitness values. */
     private final List<Double> statisticsFitnessHistory = new ArrayList<>();
     /** History of D matrix. */
     private final List<RealMatrix> statisticsDHistory = new ArrayList<>();
 
     /**
      * @param maxIterations Maximal number of iterations.
      * @param stopFitness Whether to stop if objective function value is smaller than
      * {@code stopFitness}.
      * @param isActiveCMA Chooses the covariance matrix update method.
      * @param diagonalOnly Number of initial iterations, where the covariance matrix
      * remains diagonal.
      * @param checkFeasableCount Determines how often new random objective variables are
      * generated in case they are out of bounds.
      * @param rng Random generator.
      * @param generateStatistics Whether statistic data is collected.
      * @param checker Convergence checker.
      *
      * @since 3.1
      */
     public CMAESOptimizer(int maxIterations,
                           double stopFitness,
                           boolean isActiveCMA,
                           int diagonalOnly,
                           int checkFeasableCount,
                           UniformRandomProvider rng,
                           boolean generateStatistics,
                           ConvergenceChecker<PointValuePair> checker) {
         super(checker);
         this.maxIterations = maxIterations;
         this.stopFitness = stopFitness;
         this.isActiveCMA = isActiveCMA;
         this.diagonalOnly = diagonalOnly;
         this.checkFeasableCount = Math.max(0, checkFeasableCount);
         this.random = NormalDistribution.of(0, 1).createSampler(rng);
         this.generateStatistics = generateStatistics;
     }
 
     /**
      * @return History of sigma values.
      */
     public List<Double> getStatisticsSigmaHistory() {
         return statisticsSigmaHistory;
     }
 
     /**
      * @return History of mean matrix.
      */
     public List<RealMatrix> getStatisticsMeanHistory() {
         return statisticsMeanHistory;
     }
 
     /**
      * @return History of fitness values.
      */
     public List<Double> getStatisticsFitnessHistory() {
         return statisticsFitnessHistory;
     }
 
     /**
      * @return History of D matrix.
      */
     public List<RealMatrix> getStatisticsDHistory() {
         return statisticsDHistory;
     }
 
     /**
      * {@inheritDoc}
      *
      * @param optData Optimization data. In addition to those documented in
      * {@link MultivariateOptimizer#parseOptimizationData(OptimizationData[])
      * MultivariateOptimizer}, this method will register the following data:
      * <ul>
      *  <li>
      *   {@link Sigma} values define the initial coordinate-wise standard
      *   deviations for sampling new search points around the initial guess.
      *   It is suggested to set them to the estimated distance from the
      *   initial to the desired optimum.
      *   Small values induce the search to be more local (and very small
      *   values are more likely to find a local optimum close to the initial
      *   guess).
      *   Too small values might however lead to early termination.
      *  </li>
      *  <li>
      *   {@link PopulationSize} is the number of offsprings and the primary
      *   strategy parameter.
      *   In the absence of better clues, a good default could be an integer
      *   close to {@code 4 + 3 ln(n)}, where {@code n} is the number of
      *   optimized parameters.
      *   Increasing the population size improves global search properties at
      *   the expense of speed (which in general decreases at most linearly
      *   with increasing population size).
      *  </li>
      * </ul>
      * @return {@inheritDoc}
      * @throws TooManyEvaluationsException if the maximal number of evaluations
      * is exceeded.
      * @throws DimensionMismatchException if the initial guess, target, and
      * weight arguments have inconsistent dimensions.
      */
     @Override
     public PointValuePair optimize(OptimizationData... optData) {
         // Set up base class and perform computation.
         return super.optimize(optData);
     }
 
     /** {@inheritDoc} */
     @Override
     protected PointValuePair doOptimize() {
          // -------------------- Initialization --------------------------------
         isMinimize = getGoalType().equals(GoalType.MINIMIZE);
         final FitnessFunction fitfun = new FitnessFunction();
         final double[] guess = getStartPoint();
         // number of objective variables/problem dimension
         dimension = guess.length;
         initializeCMA(guess);
         iterations = 0;
         ValuePenaltyPair valuePenalty = fitfun.value(guess);
         double bestValue = valuePenalty.value+valuePenalty.penalty;
         push(fitnessHistory, bestValue);
         PointValuePair optimum
             = new PointValuePair(getStartPoint(),
                                  isMinimize ? bestValue : -bestValue);
         PointValuePair lastResult = null;
 
         // -------------------- Generation Loop --------------------------------
         generationLoop:
         for (iterations = 1; iterations <= maxIterations; iterations++) {
             incrementIterationCount();
 
             // Generate and evaluate lambda offspring
             final RealMatrix arz = randn1(dimension, lambda);
             final RealMatrix arx = zeros(dimension, lambda);
             final double[] fitness = new double[lambda];
             final ValuePenaltyPair[] valuePenaltyPairs = new ValuePenaltyPair[lambda];
             // generate random offspring
             for (int k = 0; k < lambda; k++) {
                 RealMatrix arxk = null;
                 for (int i = 0; i <= checkFeasableCount; i++) {
                     if (diagonalOnly <= 0) {
                         arxk = xmean.add(BD.multiply(arz.getColumnMatrix(k))
                                          .scalarMultiply(sigma)); // m + sig * Normal(0,C)
                     } else {
                         arxk = xmean.add(times(diagD,arz.getColumnMatrix(k))
                                          .scalarMultiply(sigma));
                     }
                     if (i >= checkFeasableCount ||
                         fitfun.isFeasible(arxk.getColumn(0))) {
                         break;
                     }
                     // regenerate random arguments for row
                     arz.setColumn(k, randn(dimension));
                 }
                 copyColumn(arxk, 0, arx, k);
                 try {
                     valuePenaltyPairs[k] = fitfun.value(arx.getColumn(k)); // compute fitness
                 } catch (TooManyEvaluationsException e) {
                     break generationLoop;
                 }
             }
             // Compute fitnesses by adding value and penalty after scaling by value range.
             double valueRange = valueRange(valuePenaltyPairs);
             for (int iValue=0;iValue<valuePenaltyPairs.length;iValue++) {
                  fitness[iValue] = valuePenaltyPairs[iValue].value + valuePenaltyPairs[iValue].penalty*valueRange;
             }
             // Sort by fitness and compute weighted mean into xmean
             final int[] arindex = sortedIndices(fitness);
             // Calculate new xmean, this is selection and recombination
             final RealMatrix xold = xmean; // for speed up of Eq. (2) and (3)
             final RealMatrix bestArx = selectColumns(arx, Arrays.copyOf(arindex, mu));
             xmean = bestArx.multiply(weights);
             final RealMatrix bestArz = selectColumns(arz, Arrays.copyOf(arindex, mu));
             final RealMatrix zmean = bestArz.multiply(weights);
             final boolean hsig = updateEvolutionPaths(zmean, xold);
             if (diagonalOnly <= 0) {
                 updateCovariance(hsig, bestArx, arz, arindex, xold);
             } else {
                 updateCovarianceDiagonalOnly(hsig, bestArz);
             }
             // Adapt step size sigma - Eq. (5)
             sigma *= JdkMath.exp(JdkMath.min(1, (normps/chiN - 1) * cs / damps));
             final double bestFitness = fitness[arindex[0]];
             final double worstFitness = fitness[arindex[arindex.length - 1]];
             if (bestValue > bestFitness) {
                 bestValue = bestFitness;
                 lastResult = optimum;
                 optimum = new PointValuePair(fitfun.repair(bestArx.getColumn(0)),
                                              isMinimize ? bestFitness : -bestFitness);
                 if (getConvergenceChecker() != null && lastResult != null &&
                     getConvergenceChecker().converged(iterations, optimum, lastResult)) {
                     break generationLoop;
                 }
             }
             // handle termination criteria
             // Break, if fitness is good enough
             if (stopFitness != 0 && bestFitness < (isMinimize ? stopFitness : -stopFitness)) {
                 break generationLoop;
             }
             final double[] sqrtDiagC = sqrt(diagC).getColumn(0);
             final double[] pcCol = pc.getColumn(0);
             for (int i = 0; i < dimension; i++) {
                 if (sigma * JdkMath.max(JdkMath.abs(pcCol[i]), sqrtDiagC[i]) > stopTolX) {
                     break;
                 }
                 if (i >= dimension - 1) {
                     break generationLoop;
                 }
             }
             for (int i = 0; i < dimension; i++) {
                 if (sigma * sqrtDiagC[i] > stopTolUpX) {
                     break generationLoop;
                 }
             }
             final double historyBest = min(fitnessHistory);
             final double historyWorst = max(fitnessHistory);
             if (iterations > 2 &&
                 JdkMath.max(historyWorst, worstFitness) -
                 JdkMath.min(historyBest, bestFitness) < stopTolFun) {
                 break generationLoop;
             }
             if (iterations > fitnessHistory.length &&
                 historyWorst - historyBest < stopTolHistFun) {
                 break generationLoop;
             }
             // condition number of the covariance matrix exceeds 1e14
             if (max(diagD) / min(diagD) > 1e7) {
                 break generationLoop;
             }
             // user-defined termination
             if (getConvergenceChecker() != null) {
                 final PointValuePair current
                     = new PointValuePair(bestArx.getColumn(0),
                                          isMinimize ? bestFitness : -bestFitness);
                 if (lastResult != null &&
                     getConvergenceChecker().converged(iterations, current, lastResult)) {
                     break generationLoop;
                     }
                 lastResult = current;
             }
             // Adjust step size in case of equal function values (flat fitness)
             if (bestValue == fitness[arindex[(int)(0.1+lambda/4.)]]) {
                 sigma *= JdkMath.exp(0.2 + cs / damps);
             }
             if (iterations > 2 && JdkMath.max(historyWorst, bestFitness) -
                 JdkMath.min(historyBest, bestFitness) == 0) {
                 sigma *= JdkMath.exp(0.2 + cs / damps);
             }
             // store best in history
             push(fitnessHistory,bestFitness);
             if (generateStatistics) {
                 statisticsSigmaHistory.add(sigma);
                 statisticsFitnessHistory.add(bestFitness);
                 statisticsMeanHistory.add(xmean.transpose());
                 statisticsDHistory.add(diagD.transpose().scalarMultiply(1E5));
             }
         }
         return optimum;
     }
 
     /**
      * Scans the list of (required and optional) optimization data that
      * characterize the problem.
      *
      * @param optData Optimization data. The following data will be looked for:
      * <ul>
      *  <li>{@link Sigma}</li>
      *  <li>{@link PopulationSize}</li>
      * </ul>
      */
     @Override
     protected void parseOptimizationData(OptimizationData... optData) {
         // Allow base class to register its own data.
         super.parseOptimizationData(optData);
 
         // The existing values (as set by the previous call) are reused if
         // not provided in the argument list.
         for (OptimizationData data : optData) {
             if (data instanceof Sigma) {
                 inputSigma = ((Sigma) data).getSigma();
                 continue;
             }
             if (data instanceof PopulationSize) {
                 lambda = ((PopulationSize) data).getPopulationSize();
                 continue;
             }
         }
 
         checkParameters();
     }
 
     /**
      * Checks dimensions and values of boundaries and inputSigma if defined.
      */
     private void checkParameters() {
         if (inputSigma != null) {
             final double[] init = getStartPoint();
 
             if (inputSigma.length != init.length) {
                 throw new DimensionMismatchException(inputSigma.length, init.length);
             }
 
             final double[] lB = getLowerBound();
             final double[] uB = getUpperBound();
 
             for (int i = 0; i < init.length; i++) {
                 if (inputSigma[i] > uB[i] - lB[i]) {
                     throw new OutOfRangeException(inputSigma[i], 0, uB[i] - lB[i]);
                 }
             }
         }
     }
 
     /**
      * Initialization of the dynamic search parameters.
      *
      * @param guess Initial guess for the arguments of the fitness function.
      */
     private void initializeCMA(double[] guess) {
         if (lambda <= 0) {
             throw new NotStrictlyPositiveException(lambda);
         }
         // initialize sigma
         final double[][] sigmaArray = new double[guess.length][1];
         for (int i = 0; i < guess.length; i++) {
             sigmaArray[i][0] = inputSigma[i];
         }
         final RealMatrix insigma = new Array2DRowRealMatrix(sigmaArray, false);
         sigma = max(insigma); // overall standard deviation
 
         // initialize termination criteria
         stopTolUpX = 1e3 * max(insigma);
         stopTolX = 1e-11 * max(insigma);
         stopTolFun = 1e-12;
         stopTolHistFun = 1e-13;
 
         // initialize selection strategy parameters
         mu = lambda / 2; // number of parents/points for recombination
         logMu2 = JdkMath.log(mu + 0.5);
         weights = log(sequence(1, mu, 1)).scalarMultiply(-1).scalarAdd(logMu2);
         double sumw = 0;
         double sumwq = 0;
         for (int i = 0; i < mu; i++) {
             double w = weights.getEntry(i, 0);
             sumw += w;
             sumwq += w * w;
         }
         weights = weights.scalarMultiply(1 / sumw);
         mueff = sumw * sumw / sumwq; // variance-effectiveness of sum w_i x_i
 
         // initialize dynamic strategy parameters and constants
         cc = (4 + mueff / dimension) /
                 (dimension + 4 + 2 * mueff / dimension);
         cs = (mueff + 2) / (dimension + mueff + 3.);
         damps = (1 + 2 * JdkMath.max(0, JdkMath.sqrt((mueff - 1) /
                                                        (dimension + 1)) - 1)) *
             JdkMath.max(0.3,
                          1 - dimension / (1e-6 + maxIterations)) + cs; // minor increment
         ccov1 = 2 / ((dimension + 1.3) * (dimension + 1.3) + mueff);
         ccovmu = JdkMath.min(1 - ccov1, 2 * (mueff - 2 + 1 / mueff) /
                               ((dimension + 2) * (dimension + 2) + mueff));
         ccov1Sep = JdkMath.min(1, ccov1 * (dimension + 1.5) / 3);
         ccovmuSep = JdkMath.min(1 - ccov1, ccovmu * (dimension + 1.5) / 3);
         chiN = JdkMath.sqrt(dimension) *
                 (1 - 1 / ((double) 4 * dimension) + 1 / ((double) 21 * dimension * dimension));
         // initialize CMA internal values - updated each generation
         xmean = MatrixUtils.createColumnRealMatrix(guess); // objective variables
         diagD = insigma.scalarMultiply(1 / sigma);
         diagC = square(diagD);
         pc = zeros(dimension, 1); // evolution paths for C and sigma
         ps = zeros(dimension, 1); // B defines the coordinate system
         normps = ps.getFrobeniusNorm();
 
         B = eye(dimension, dimension);
         D = ones(dimension, 1); // diagonal D defines the scaling
         BD = times(B, repmat(diagD.transpose(), dimension, 1));
         C = B.multiply(diag(square(D)).multiply(B.transpose())); // covariance
         historySize = 10 + (int) (3 * 10 * dimension / (double) lambda);
         fitnessHistory = new double[historySize]; // history of fitness values
         for (int i = 0; i < historySize; i++) {
             fitnessHistory[i] = Double.MAX_VALUE;
         }
     }
 
     /**
      * Update of the evolution paths ps and pc.
      *
      * @param zmean Weighted row matrix of the gaussian random numbers generating
      * the current offspring.
      * @param xold xmean matrix of the previous generation.
      * @return hsig flag indicating a small correction.
      */
     private boolean updateEvolutionPaths(RealMatrix zmean, RealMatrix xold) {
         ps = ps.scalarMultiply(1 - cs).add(
                 B.multiply(zmean).scalarMultiply(
                         JdkMath.sqrt(cs * (2 - cs) * mueff)));
         normps = ps.getFrobeniusNorm();
         final boolean hsig = normps /
             JdkMath.sqrt(1 - JdkMath.pow(1 - cs, 2 * iterations)) /
             chiN < 1.4 + 2 / ((double) dimension + 1);
         pc = pc.scalarMultiply(1 - cc);
         if (hsig) {
             pc = pc.add(xmean.subtract(xold).scalarMultiply(JdkMath.sqrt(cc * (2 - cc) * mueff) / sigma));
         }
         return hsig;
     }
 
     /**
      * Update of the covariance matrix C for diagonalOnly > 0.
      *
      * @param hsig Flag indicating a small correction.
      * @param bestArz Fitness-sorted matrix of the gaussian random values of the
      * current offspring.
      */
     private void updateCovarianceDiagonalOnly(boolean hsig,
                                               final RealMatrix bestArz) {
         // minor correction if hsig==false
         double oldFac = hsig ? 0 : ccov1Sep * cc * (2 - cc);
         oldFac += 1 - ccov1Sep - ccovmuSep;
         diagC = diagC.scalarMultiply(oldFac) // regard old matrix
             .add(square(pc).scalarMultiply(ccov1Sep)) // plus rank one update
             .add(times(diagC, square(bestArz).multiply(weights)) // plus rank mu update
                  .scalarMultiply(ccovmuSep));
         diagD = sqrt(diagC); // replaces eig(C)
         if (diagonalOnly > 1 &&
             iterations > diagonalOnly) {
             // full covariance matrix from now on
             diagonalOnly = 0;
             B = eye(dimension, dimension);
             BD = diag(diagD);
             C = diag(diagC);
         }
     }
 
     /**
      * Update of the covariance matrix C.
      *
      * @param hsig Flag indicating a small correction.
      * @param bestArx Fitness-sorted matrix of the argument vectors producing the
      * current offspring.
      * @param arz Unsorted matrix containing the gaussian random values of the
      * current offspring.
      * @param arindex Indices indicating the fitness-order of the current offspring.
      * @param xold xmean matrix of the previous generation.
      */
     private void updateCovariance(boolean hsig, final RealMatrix bestArx,
                                   final RealMatrix arz, final int[] arindex,
                                   final RealMatrix xold) {
         double negccov = 0;
         if (ccov1 + ccovmu > 0) {
             final RealMatrix arpos = bestArx.subtract(repmat(xold, 1, mu))
                 .scalarMultiply(1 / sigma); // mu difference vectors
             final RealMatrix roneu = pc.multiply(pc.transpose())
                 .scalarMultiply(ccov1); // rank one update
             // minor correction if hsig==false
             double oldFac = hsig ? 0 : ccov1 * cc * (2 - cc);
             oldFac += 1 - ccov1 - ccovmu;
             if (isActiveCMA) {
                 // Adapt covariance matrix C active CMA
                 negccov = (1 - ccovmu) * 0.25 * mueff /
                     (JdkMath.pow(dimension + 2.0, 1.5) + 2 * mueff);
                 // keep at least 0.66 in all directions, small popsize are most
                 // critical
                 final double negminresidualvariance = 0.66;
                 // where to make up for the variance loss
                 final double negalphaold = 0.5;
                 // prepare vectors, compute negative updating matrix Cneg
                 final int[] arReverseIndex = reverse(arindex);
                 RealMatrix arzneg = selectColumns(arz, Arrays.copyOf(arReverseIndex, mu));
                 RealMatrix arnorms = sqrt(sumRows(square(arzneg)));
                 final int[] idxnorms = sortedIndices(arnorms.getRow(0));
                 final RealMatrix arnormsSorted = selectColumns(arnorms, idxnorms);
                 final int[] idxReverse = reverse(idxnorms);
                 final RealMatrix arnormsReverse = selectColumns(arnorms, idxReverse);
                 arnorms = divide(arnormsReverse, arnormsSorted);
                 final int[] idxInv = inverse(idxnorms);
                 final RealMatrix arnormsInv = selectColumns(arnorms, idxInv);
                 // check and set learning rate negccov
                 final double negcovMax = (1 - negminresidualvariance) /
                     square(arnormsInv).multiply(weights).getEntry(0, 0);
                 if (negccov > negcovMax) {
                     negccov = negcovMax;
                 }
                 arzneg = times(arzneg, repmat(arnormsInv, dimension, 1));
                 final RealMatrix artmp = BD.multiply(arzneg);
                 final RealMatrix cNeg = artmp.multiply(diag(weights)).multiply(artmp.transpose());
                 oldFac += negalphaold * negccov;
                 C = C.scalarMultiply(oldFac)
                     .add(roneu) // regard old matrix
                     .add(arpos.scalarMultiply( // plus rank one update
                                               ccovmu + (1 - negalphaold) * negccov) // plus rank mu update
                          .multiply(times(repmat(weights, 1, dimension),
                                          arpos.transpose())))
                     .subtract(cNeg.scalarMultiply(negccov));
             } else {
                 // Adapt covariance matrix C - nonactive
                 C = C.scalarMultiply(oldFac) // regard old matrix
                     .add(roneu) // plus rank one update
                     .add(arpos.scalarMultiply(ccovmu) // plus rank mu update
                          .multiply(times(repmat(weights, 1, dimension),
                                          arpos.transpose())));
             }
         }
         updateBD(negccov);
     }
 
     /**
      * Update B and D from C.
      *
      * @param negccov Negative covariance factor.
      */
     private void updateBD(double negccov) {
         if (ccov1 + ccovmu + negccov > 0 &&
             (iterations % 1. / (ccov1 + ccovmu + negccov) / dimension / 10.) < 1) {
             // to achieve O(N^2)
             C = triu(C, 0).add(triu(C, 1).transpose());
             // enforce symmetry to prevent complex numbers
             final EigenDecomposition eig = new EigenDecomposition(C);
             B = eig.getV(); // eigen decomposition, B==normalized eigenvectors
             D = eig.getD();
             diagD = diag(D);
             if (min(diagD) <= 0) {
                 for (int i = 0; i < dimension; i++) {
                     if (diagD.getEntry(i, 0) < 0) {
                         diagD.setEntry(i, 0, 0);
                     }
                 }
                 final double tfac = max(diagD) / 1e14;
                 C = C.add(eye(dimension, dimension).scalarMultiply(tfac));
                 diagD = diagD.add(ones(dimension, 1).scalarMultiply(tfac));
             }
             if (max(diagD) > 1e14 * min(diagD)) {
                 final double tfac = max(diagD) / 1e14 - min(diagD);
                 C = C.add(eye(dimension, dimension).scalarMultiply(tfac));
                 diagD = diagD.add(ones(dimension, 1).scalarMultiply(tfac));
             }
             diagC = diag(C);
             diagD = sqrt(diagD); // D contains standard deviations now
             BD = times(B, repmat(diagD.transpose(), dimension, 1)); // O(n^2)
         }
     }
 
     /**
      * Pushes the current best fitness value in a history queue.
      *
      * @param vals History queue.
      * @param val Current best fitness value.
      */
     private static void push(double[] vals, double val) {
         if (vals.length - 1 > 0) {
             System.arraycopy(vals, 0, vals, 1, vals.length - 1);
         }
         vals[0] = val;
     }
 
     /**
      * Sorts fitness values.
      *
      * @param doubles Array of values to be sorted.
      * @return a sorted array of indices pointing into doubles.
      */
     private int[] sortedIndices(final double[] doubles) {
         final DoubleIndex[] dis = new DoubleIndex[doubles.length];
         for (int i = 0; i < doubles.length; i++) {
             dis[i] = new DoubleIndex(doubles[i], i);
         }
         Arrays.sort(dis);
         final int[] indices = new int[doubles.length];
         for (int i = 0; i < doubles.length; i++) {
             indices[i] = dis[i].index;
         }
         return indices;
     }
    /**
      * Get range of values.
      *
      * @param vpPairs Array of valuePenaltyPairs to get range from.
      * @return a double equal to maximum value minus minimum value.
      */
     private double valueRange(final ValuePenaltyPair[] vpPairs) {
         double max = Double.NEGATIVE_INFINITY;
         double min = Double.MAX_VALUE;
         for (ValuePenaltyPair vpPair:vpPairs) {
             if (vpPair.value > max) {
                 max = vpPair.value;
             }
             if (vpPair.value < min) {
                 min = vpPair.value;
             }
         }
         return max-min;
     }
 
     /**
      * Used to sort fitness values. Sorting is always in lower value first
      * order.
      */
     private static final class DoubleIndex implements Comparable<DoubleIndex> {
         /** Value to compare. */
         private final double value;
         /** Index into sorted array. */
         private final int index;
 
         /**
          * @param value Value to compare.
          * @param index Index into sorted array.
          */
         DoubleIndex(double value, int index) {
             this.value = value;
             this.index = index;
         }
 
         /** {@inheritDoc} */
         @Override
         public int compareTo(DoubleIndex o) {
             return Double.compare(value, o.value);
         }
 
         /** {@inheritDoc} */
         @Override
         public boolean equals(Object other) {
 
             if (this == other) {
                 return true;
             }
 
             if (other instanceof DoubleIndex) {
                 return Double.compare(value, ((DoubleIndex) other).value) == 0;
             }
 
             return false;
         }
 
         /** {@inheritDoc} */
         @Override
         public int hashCode() {
             long bits = Double.doubleToLongBits(value);
             return (int) (1438542 ^ (bits >>> 32) ^ bits);
         }
     }
     /**
      * Stores the value and penalty (for repair of out of bounds point).
      */
     private static final class ValuePenaltyPair {
         /** Objective function value. */
         private double value;
         /** Penalty value for repair of out out of bounds points. */
         private double penalty;
 
         /**
          * @param value Function value.
          * @param penalty Out-of-bounds penalty.
         */
         ValuePenaltyPair(final double value, final double penalty) {
             this.value   = value;
             this.penalty = penalty;
         }
     }
 
 
     /**
      * Normalizes fitness values to the range [0,1]. Adds a penalty to the
      * fitness value if out of range.
      */
     private final class FitnessFunction {
         /**
          * Flag indicating whether the objective variables are forced into their
          * bounds if defined.
          */
         private final boolean isRepairMode;
 
         /** Simple constructor.
          */
         FitnessFunction() {
             isRepairMode = true;
         }
 
         /**
          * @param point Normalized objective variables.
          * @return the objective value + penalty for violated bounds.
          */
         public ValuePenaltyPair value(final double[] point) {
             final MultivariateFunction func = CMAESOptimizer.this.getObjectiveFunction();
             double value;
             double penalty = 0;
             if (isRepairMode) {
                 double[] repaired = repair(point);
                 value = func.value(repaired);
                 penalty = penalty(point, repaired);
             } else {
                 value = func.value(point);
             }
             value = isMinimize ? value : -value;
             penalty = isMinimize ? penalty : -penalty;
             return new ValuePenaltyPair(value,penalty);
         }
 
         /**
          * @param x Normalized objective variables.
          * @return {@code true} if in bounds.
          */
         public boolean isFeasible(final double[] x) {
             final double[] lB = CMAESOptimizer.this.getLowerBound();
             final double[] uB = CMAESOptimizer.this.getUpperBound();
 
             for (int i = 0; i < x.length; i++) {
                 if (x[i] < lB[i]) {
                     return false;
                 }
                 if (x[i] > uB[i]) {
                     return false;
                 }
             }
             return true;
         }
 
         /**
          * @param x Normalized objective variables.
          * @return the repaired (i.e. all in bounds) objective variables.
          */
         private double[] repair(final double[] x) {
             final double[] lB = CMAESOptimizer.this.getLowerBound();
             final double[] uB = CMAESOptimizer.this.getUpperBound();
 
             final double[] repaired = new double[x.length];
             for (int i = 0; i < x.length; i++) {
                 if (x[i] < lB[i]) {
                     repaired[i] = lB[i];
                 } else if (x[i] > uB[i]) {
                     repaired[i] = uB[i];
                 } else {
                     repaired[i] = x[i];
                 }
             }
             return repaired;
         }
 
         /**
          * @param x Normalized objective variables.
          * @param repaired Repaired objective variables.
          * @return Penalty value according to the violation of the bounds.
          */
         private double penalty(final double[] x, final double[] repaired) {
             double penalty = 0;
             for (int i = 0; i < x.length; i++) {
                 double diff = JdkMath.abs(x[i] - repaired[i]);
                 penalty += diff;
             }
             return isMinimize ? penalty : -penalty;
         }
     }
 
     // -----Matrix utility functions similar to the Matlab build in functions------
 
     /**
      * @param m Input matrix
      * @return Matrix representing the element-wise logarithm of m.
      */
     private static RealMatrix log(final RealMatrix m) {
         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 d[r][c] = JdkMath.log(m.getEntry(r, c));
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @return Matrix representing the element-wise square root of m.
      */
     private static RealMatrix sqrt(final RealMatrix m) {
         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 d[r][c] = JdkMath.sqrt(m.getEntry(r, c));
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @return Matrix representing the element-wise square of m.
      */
     private static RealMatrix square(final RealMatrix m) {
         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 double e = m.getEntry(r, c);
                 d[r][c] = e * e;
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix 1.
      * @param n Input matrix 2.
      * @return the matrix where the elements of m and n are element-wise multiplied.
      */
     private static RealMatrix times(final RealMatrix m, final RealMatrix n) {
         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 d[r][c] = m.getEntry(r, c) * n.getEntry(r, c);
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix 1.
      * @param n Input matrix 2.
      * @return Matrix where the elements of m and n are element-wise divided.
      */
     private static RealMatrix divide(final RealMatrix m, final RealMatrix n) {
         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 d[r][c] = m.getEntry(r, c) / n.getEntry(r, c);
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @param cols Columns to select.
      * @return Matrix representing the selected columns.
      */
     private static RealMatrix selectColumns(final RealMatrix m, final int[] cols) {
         final double[][] d = new double[m.getRowDimension()][cols.length];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < cols.length; c++) {
                 d[r][c] = m.getEntry(r, cols[c]);
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @param k Diagonal position.
      * @return Upper triangular part of matrix.
      */
     private static RealMatrix triu(final RealMatrix m, int k) {
         final double[][] d = new double[m.getRowDimension()][m.getColumnDimension()];
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 d[r][c] = r <= c - k ? m.getEntry(r, c) : 0;
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @return Row matrix representing the sums of the rows.
      */
     private static RealMatrix sumRows(final RealMatrix m) {
         final double[][] d = new double[1][m.getColumnDimension()];
         for (int c = 0; c < m.getColumnDimension(); c++) {
             double sum = 0;
             for (int r = 0; r < m.getRowDimension(); r++) {
                 sum += m.getEntry(r, c);
             }
             d[0][c] = sum;
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @return the diagonal n-by-n matrix if m is a column matrix or the column
      * matrix representing the diagonal if m is a n-by-n matrix.
      */
     private static RealMatrix diag(final RealMatrix m) {
         if (m.getColumnDimension() == 1) {
             final double[][] d = new double[m.getRowDimension()][m.getRowDimension()];
             for (int i = 0; i < m.getRowDimension(); i++) {
                 d[i][i] = m.getEntry(i, 0);
             }
             return new Array2DRowRealMatrix(d, false);
         } else {
             final double[][] d = new double[m.getRowDimension()][1];
             for (int i = 0; i < m.getColumnDimension(); i++) {
                 d[i][0] = m.getEntry(i, i);
             }
             return new Array2DRowRealMatrix(d, false);
         }
     }
 
     /**
      * Copies a column from m1 to m2.
      *
      * @param m1 Source matrix.
      * @param col1 Source column.
      * @param m2 Target matrix.
      * @param col2 Target column.
      */
     private static void copyColumn(final RealMatrix m1, int col1,
                                    RealMatrix m2, int col2) {
         for (int i = 0; i < m1.getRowDimension(); i++) {
             m2.setEntry(i, col2, m1.getEntry(i, col1));
         }
     }
 
     /**
      * @param n Number of rows.
      * @param m Number of columns.
      * @return n-by-m matrix filled with 1.
      */
     private static RealMatrix ones(int n, int m) {
         final double[][] d = new double[n][m];
         for (int r = 0; r < n; r++) {
             Arrays.fill(d[r], 1);
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param n Number of rows.
      * @param m Number of columns.
      * @return n-by-m matrix of 0 values out of diagonal, and 1 values on
      * the diagonal.
      */
     private static RealMatrix eye(int n, int m) {
         final double[][] d = new double[n][m];
         for (int r = 0; r < n; r++) {
             if (r < m) {
                 d[r][r] = 1;
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param n Number of rows.
      * @param m Number of columns.
      * @return n-by-m matrix of zero values.
      */
     private static RealMatrix zeros(int n, int m) {
         return new Array2DRowRealMatrix(n, m);
     }
 
     /**
      * @param mat Input matrix.
      * @param n Number of row replicates.
      * @param m Number of column replicates.
      * @return a matrix which replicates the input matrix in both directions.
      */
     private static RealMatrix repmat(final RealMatrix mat, int n, int m) {
         final int rd = mat.getRowDimension();
         final int cd = mat.getColumnDimension();
         final double[][] d = new double[n * rd][m * cd];
         for (int r = 0; r < n * rd; r++) {
             for (int c = 0; c < m * cd; c++) {
                 d[r][c] = mat.getEntry(r % rd, c % cd);
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param start Start value.
      * @param end End value.
      * @param step Step size.
      * @return a sequence as column matrix.
      */
     private static RealMatrix sequence(double start, double end, double step) {
         final int size = (int) ((end - start) / step + 1);
         final double[][] d = new double[size][1];
         double value = start;
         for (int r = 0; r < size; r++) {
             d[r][0] = value;
             value += step;
         }
         return new Array2DRowRealMatrix(d, false);
     }
 
     /**
      * @param m Input matrix.
      * @return the maximum of the matrix element values.
      */
     private static double max(final RealMatrix m) {
         double max = -Double.MAX_VALUE;
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 double e = m.getEntry(r, c);
                 if (max < e) {
                     max = e;
                 }
             }
         }
         return max;
     }
 
     /**
      * @param m Input matrix.
      * @return the minimum of the matrix element values.
      */
     private static double min(final RealMatrix m) {
         double min = Double.MAX_VALUE;
         for (int r = 0; r < m.getRowDimension(); r++) {
             for (int c = 0; c < m.getColumnDimension(); c++) {
                 double e = m.getEntry(r, c);
                 if (min > e) {
                     min = e;
                 }
             }
         }
         return min;
     }
 
     /**
      * @param m Input array.
      * @return the maximum of the array values.
      */
     private static double max(final double[] m) {
         double max = -Double.MAX_VALUE;
         for (int r = 0; r < m.length; r++) {
             if (max < m[r]) {
                 max = m[r];
             }
         }
         return max;
     }
 
     /**
      * @param m Input array.
      * @return the minimum of the array values.
      */
     private static double min(final double[] m) {
         double min = Double.MAX_VALUE;
         for (int r = 0; r < m.length; r++) {
             if (min > m[r]) {
                 min = m[r];
             }
         }
         return min;
     }
 
     /**
      * @param indices Input index array.
      * @return the inverse of the mapping defined by indices.
      */
     private static int[] inverse(final int[] indices) {
         final int[] inverse = new int[indices.length];
         for (int i = 0; i < indices.length; i++) {
             inverse[indices[i]] = i;
         }
         return inverse;
     }
 
     /**
      * @param indices Input index array.
      * @return the indices in inverse order (last is first).
      */
     private static int[] reverse(final int[] indices) {
         final int[] reverse = new int[indices.length];
         for (int i = 0; i < indices.length; i++) {
             reverse[i] = indices[indices.length - i - 1];
         }
         return reverse;
     }
 
     /**
      * @param size Length of random array.
      * @return an array of Gaussian random numbers.
      */
     private double[] randn(int size) {
         final double[] randn = new double[size];
         for (int i = 0; i < size; i++) {
             randn[i] = random.sample();
         }
         return randn;
     }
 
     /**
      * @param size Number of rows.
      * @param popSize Population size.
      * @return a 2-dimensional matrix of Gaussian random numbers.
      */
     private RealMatrix randn1(int size, int popSize) {
         final double[][] d = new double[size][popSize];
         for (int r = 0; r < size; r++) {
             for (int c = 0; c < popSize; c++) {
                 d[r][c] = random.sample();
             }
         }
         return new Array2DRowRealMatrix(d, false);
     }
 }