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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* 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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* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
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package org.apache.commons.math3.optimization.direct;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import org.apache.commons.math3.analysis.MultivariateFunction;
import org.apache.commons.math3.exception.DimensionMismatchException;
import org.apache.commons.math3.exception.NotPositiveException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.OutOfRangeException;
import org.apache.commons.math3.exception.TooManyEvaluationsException;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.EigenDecomposition;
import org.apache.commons.math3.linear.MatrixUtils;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.optimization.ConvergenceChecker;
import org.apache.commons.math3.optimization.OptimizationData;
import org.apache.commons.math3.optimization.GoalType;
import org.apache.commons.math3.optimization.MultivariateOptimizer;
import org.apache.commons.math3.optimization.PointValuePair;
import org.apache.commons.math3.optimization.SimpleValueChecker;
import org.apache.commons.math3.random.MersenneTwister;
import org.apache.commons.math3.random.RandomGenerator;
import org.apache.commons.math3.util.FastMath;
import org.apache.commons.math3.util.MathArrays;
/**
* An implementation of the active Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
* for non-linear, non-convex, non-smooth, global function minimization.
* 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.
*
* 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 variables also the derivative-free simplex
* direct search method (Nelder and Mead) can be faster, but it is
* far less reliable than CMA-ES.
*
* 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
* function evaluations. On difficult problems the complete
* optimization (a single run) is expected to take roughly between
* and
* function evaluations.
*
* 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.
*
* For more information, please refer to the following links:
*
*
* @deprecated As of 3.1 (to be removed in 4.0).
* @since 3.0
*/
@Deprecated
public class CMAESOptimizer
extends BaseAbstractMultivariateSimpleBoundsOptimizer
implements MultivariateOptimizer {
/** Default value for {@link #checkFeasableCount}: {@value}. */
public static final int DEFAULT_CHECKFEASABLECOUNT = 0;
/** Default value for {@link #stopFitness}: {@value}. */
public static final double DEFAULT_STOPFITNESS = 0;
/** Default value for {@link #isActiveCMA}: {@value}. */
public static final boolean DEFAULT_ISACTIVECMA = true;
/** Default value for {@link #maxIterations}: {@value}. */
public static final int DEFAULT_MAXITERATIONS = 30000;
/** Default value for {@link #diagonalOnly}: {@value}. */
public static final int DEFAULT_DIAGONALONLY = 0;
/** Default value for {@link #random}. */
public static final RandomGenerator DEFAULT_RANDOMGENERATOR = new MersenneTwister();
// 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 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 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 Simple Modification in CMA-ES
*/
private int diagonalOnly = 0;
/** Number of objective variables/problem dimension */
private boolean isMinimize = true;
/** Indicates whether statistic data is collected. */
private boolean generateStatistics = false;
// termination criteria
/** Maximal number of iterations allowed. */
private int maxIterations;
/** Limit for fitness value. */
private 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;
/** Random generator. */
private RandomGenerator random;
/** History of sigma values. */
private List statisticsSigmaHistory = new ArrayList();
/** History of mean matrix. */
private List statisticsMeanHistory = new ArrayList();
/** History of fitness values. */
private List statisticsFitnessHistory = new ArrayList();
/** History of D matrix. */
private List statisticsDHistory = new ArrayList();
/**
* Default constructor, uses default parameters
*
* @deprecated As of version 3.1: Parameter {@code lambda} must be
* passed with the call to {@link #optimize(int,MultivariateFunction,GoalType,OptimizationData[])
* optimize} (whereas in the current code it is set to an undocumented value).
*/
@Deprecated
public CMAESOptimizer() {
this(0);
}
/**
* @param lambda Population size.
* @deprecated As of version 3.1: Parameter {@code lambda} must be
* passed with the call to {@link #optimize(int,MultivariateFunction,GoalType,OptimizationData[])
* optimize} (whereas in the current code it is set to an undocumented value)..
*/
@Deprecated
public CMAESOptimizer(int lambda) {
this(lambda, null, DEFAULT_MAXITERATIONS, DEFAULT_STOPFITNESS,
DEFAULT_ISACTIVECMA, DEFAULT_DIAGONALONLY,
DEFAULT_CHECKFEASABLECOUNT, DEFAULT_RANDOMGENERATOR,
false, null);
}
/**
* @param lambda Population size.
* @param inputSigma Initial standard deviations to sample new points
* around the initial guess.
* @deprecated As of version 3.1: Parameters {@code lambda} and {@code inputSigma} must be
* passed with the call to {@link #optimize(int,MultivariateFunction,GoalType,OptimizationData[])
* optimize}.
*/
@Deprecated
public CMAESOptimizer(int lambda, double[] inputSigma) {
this(lambda, inputSigma, DEFAULT_MAXITERATIONS, DEFAULT_STOPFITNESS,
DEFAULT_ISACTIVECMA, DEFAULT_DIAGONALONLY,
DEFAULT_CHECKFEASABLECOUNT, DEFAULT_RANDOMGENERATOR, false);
}
/**
* @param lambda Population size.
* @param inputSigma Initial standard deviations to sample new points
* around the initial guess.
* @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 random Random generator.
* @param generateStatistics Whether statistic data is collected.
* @deprecated See {@link SimpleValueChecker#SimpleValueChecker()}
*/
@Deprecated
public CMAESOptimizer(int lambda, double[] inputSigma,
int maxIterations, double stopFitness,
boolean isActiveCMA, int diagonalOnly, int checkFeasableCount,
RandomGenerator random, boolean generateStatistics) {
this(lambda, inputSigma, maxIterations, stopFitness, isActiveCMA,
diagonalOnly, checkFeasableCount, random, generateStatistics,
new SimpleValueChecker());
}
/**
* @param lambda Population size.
* @param inputSigma Initial standard deviations to sample new points
* around the initial guess.
* @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 random Random generator.
* @param generateStatistics Whether statistic data is collected.
* @param checker Convergence checker.
* @deprecated As of version 3.1: Parameters {@code lambda} and {@code inputSigma} must be
* passed with the call to {@link #optimize(int,MultivariateFunction,GoalType,OptimizationData[])
* optimize}.
*/
@Deprecated
public CMAESOptimizer(int lambda, double[] inputSigma,
int maxIterations, double stopFitness,
boolean isActiveCMA, int diagonalOnly, int checkFeasableCount,
RandomGenerator random, boolean generateStatistics,
ConvergenceChecker checker) {
super(checker);
this.lambda = lambda;
this.inputSigma = inputSigma == null ? null : (double[]) inputSigma.clone();
this.maxIterations = maxIterations;
this.stopFitness = stopFitness;
this.isActiveCMA = isActiveCMA;
this.diagonalOnly = diagonalOnly;
this.checkFeasableCount = checkFeasableCount;
this.random = random;
this.generateStatistics = generateStatistics;
}
/**
* @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 random 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,
RandomGenerator random,
boolean generateStatistics,
ConvergenceChecker checker) {
super(checker);
this.maxIterations = maxIterations;
this.stopFitness = stopFitness;
this.isActiveCMA = isActiveCMA;
this.diagonalOnly = diagonalOnly;
this.checkFeasableCount = checkFeasableCount;
this.random = random;
this.generateStatistics = generateStatistics;
}
/**
* @return History of sigma values.
*/
public List getStatisticsSigmaHistory() {
return statisticsSigmaHistory;
}
/**
* @return History of mean matrix.
*/
public List getStatisticsMeanHistory() {
return statisticsMeanHistory;
}
/**
* @return History of fitness values.
*/
public List getStatisticsFitnessHistory() {
return statisticsFitnessHistory;
}
/**
* @return History of D matrix.
*/
public List getStatisticsDHistory() {
return statisticsDHistory;
}
/**
* Input sigma values.
* They 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.
* @since 3.1
*/
public static class Sigma implements OptimizationData {
/** Sigma values. */
private final double[] sigma;
/**
* @param s Sigma values.
* @throws NotPositiveException if any of the array entries is smaller
* than zero.
*/
public Sigma(double[] s)
throws NotPositiveException {
for (int i = 0; i < s.length; i++) {
if (s[i] < 0) {
throw new NotPositiveException(s[i]);
}
}
sigma = s.clone();
}
/**
* @return the sigma values.
*/
public double[] getSigma() {
return sigma.clone();
}
}
/**
* Population size.
* The number of offspring is 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).
* @since 3.1
*/
public static class PopulationSize implements OptimizationData {
/** Population size. */
private final int lambda;
/**
* @param size Population size.
* @throws NotStrictlyPositiveException if {@code size <= 0}.
*/
public PopulationSize(int size)
throws NotStrictlyPositiveException {
if (size <= 0) {
throw new NotStrictlyPositiveException(size);
}
lambda = size;
}
/**
* @return the population size.
*/
public int getPopulationSize() {
return lambda;
}
}
/**
* Optimize an objective function.
*
* @param maxEval Allowed number of evaluations of the objective function.
* @param f Objective function.
* @param goalType Optimization type.
* @param optData Optimization data. The following data will be looked for:
*
* - {@link org.apache.commons.math3.optimization.InitialGuess InitialGuess}
* - {@link Sigma}
* - {@link PopulationSize}
*
* @return the point/value pair giving the optimal value for objective
* function.
*/
@Override
protected PointValuePair optimizeInternal(int maxEval, MultivariateFunction f,
GoalType goalType,
OptimizationData... optData) {
// Scan "optData" for the input specific to this optimizer.
parseOptimizationData(optData);
// The parent's method will retrieve the common parameters from
// "optData" and call "doOptimize".
return super.optimizeInternal(maxEval, f, goalType, optData);
}
/** {@inheritDoc} */
@Override
protected PointValuePair doOptimize() {
checkParameters();
// -------------------- 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;
double bestValue = fitfun.value(guess);
push(fitnessHistory, bestValue);
PointValuePair optimum = new PointValuePair(getStartPoint(),
isMinimize ? bestValue : -bestValue);
PointValuePair lastResult = null;
// -------------------- Generation Loop --------------------------------
generationLoop:
for (iterations = 1; iterations <= maxIterations; iterations++) {
// Generate and evaluate lambda offspring
final RealMatrix arz = randn1(dimension, lambda);
final RealMatrix arx = zeros(dimension, lambda);
final double[] fitness = new double[lambda];
// generate random offspring
for (int k = 0; k < lambda; k++) {
RealMatrix arxk = null;
for (int i = 0; i < checkFeasableCount + 1; 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 {
fitness[k] = fitfun.value(arx.getColumn(k)); // compute fitness
} catch (TooManyEvaluationsException e) {
break generationLoop;
}
}
// 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, MathArrays.copyOf(arindex, mu));
xmean = bestArx.multiply(weights);
final RealMatrix bestArz = selectColumns(arz, MathArrays.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 *= FastMath.exp(FastMath.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 * FastMath.max(FastMath.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 &&
FastMath.max(historyWorst, worstFitness) -
FastMath.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 *= FastMath.exp(0.2 + cs / damps);
}
if (iterations > 2 && FastMath.max(historyWorst, bestFitness) -
FastMath.min(historyBest, bestFitness) == 0) {
sigma *= FastMath.exp(0.2 + cs / damps);
}
// store best in history
push(fitnessHistory,bestFitness);
fitfun.setValueRange(worstFitness-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:
*
* - {@link Sigma}
* - {@link PopulationSize}
*
*/
private void parseOptimizationData(OptimizationData... 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;
}
}
}
/**
* Checks dimensions and values of boundaries and inputSigma if defined.
*/
private void checkParameters() {
final double[] init = getStartPoint();
final double[] lB = getLowerBound();
final double[] uB = getUpperBound();
if (inputSigma != null) {
if (inputSigma.length != init.length) {
throw new DimensionMismatchException(inputSigma.length, init.length);
}
for (int i = 0; i < init.length; i++) {
if (inputSigma[i] < 0) {
// XXX Remove this block in 4.0 (check performed in "Sigma" class).
throw new NotPositiveException(inputSigma[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) {
// XXX Line below to replace the current one in 4.0 (MATH-879).
// throw new NotStrictlyPositiveException(lambda);
lambda = 4 + (int) (3 * FastMath.log(dimension));
}
// initialize sigma
final double[][] sigmaArray = new double[guess.length][1];
for (int i = 0; i < guess.length; i++) {
// XXX Line below to replace the current one in 4.0 (MATH-868).
// sigmaArray[i][0] = inputSigma[i];
sigmaArray[i][0] = inputSigma == null ? 0.3 : 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 = FastMath.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 * FastMath.max(0, FastMath.sqrt((mueff - 1) /
(dimension + 1)) - 1)) *
FastMath.max(0.3,
1 - dimension / (1e-6 + maxIterations)) + cs; // minor increment
ccov1 = 2 / ((dimension + 1.3) * (dimension + 1.3) + mueff);
ccovmu = FastMath.min(1 - ccov1, 2 * (mueff - 2 + 1 / mueff) /
((dimension + 2) * (dimension + 2) + mueff));
ccov1Sep = FastMath.min(1, ccov1 * (dimension + 1.5) / 3);
ccovmuSep = FastMath.min(1 - ccov1, ccovmu * (dimension + 1.5) / 3);
chiN = FastMath.sqrt(dimension) *
(1 - 1 / ((double) 4 * dimension) + 1 / ((double) 21 * dimension * dimension));
// intialize 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(FastMath.sqrt(cs * (2 - cs) * mueff)));
normps = ps.getFrobeniusNorm();
final boolean hsig = normps /
FastMath.sqrt(1 - FastMath.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(FastMath.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 / (FastMath.pow(dimension + 2, 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, MathArrays.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) {
for (int i = vals.length-1; i > 0; i--) {
vals[i] = vals[i-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;
}
/**
* Used to sort fitness values. Sorting is always in lower value first
* order.
*/
private static class DoubleIndex implements Comparable {
/** 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} */
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) & 0xffffffff);
}
}
/**
* Normalizes fitness values to the range [0,1]. Adds a penalty to the
* fitness value if out of range. The penalty is adjusted by calling
* setValueRange().
*/
private class FitnessFunction {
/** Determines the penalty for boundary violations */
private double valueRange;
/**
* Flag indicating whether the objective variables are forced into their
* bounds if defined
*/
private final boolean isRepairMode;
/** Simple constructor.
*/
public FitnessFunction() {
valueRange = 1;
isRepairMode = true;
}
/**
* @param point Normalized objective variables.
* @return the objective value + penalty for violated bounds.
*/
public double value(final double[] point) {
double value;
if (isRepairMode) {
double[] repaired = repair(point);
value = CMAESOptimizer.this.computeObjectiveValue(repaired) +
penalty(point, repaired);
} else {
value = CMAESOptimizer.this.computeObjectiveValue(point);
}
return isMinimize ? value : -value;
}
/**
* @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 valueRange Adjusts the penalty computation.
*/
public void setValueRange(double valueRange) {
this.valueRange = valueRange;
}
/**
* @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 = FastMath.abs(x[i] - repaired[i]);
penalty += diff * valueRange;
}
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] = FastMath.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] = FastMath.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.nextGaussian();
}
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.nextGaussian();
}
}
return new Array2DRowRealMatrix(d, false);
}
}