public class CMAESOptimizer extends MultivariateOptimizer
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 N=10 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 30 N function evaluations. On difficult problems the complete optimization (a single run) is expected to take roughly between 30 N and 300 N2 function evaluations.

This implementation is translated and adapted from the Matlab version of the CMA-ES algorithm as implemented in module cmaes.m version 3.51.

For more information, please refer to the following links:

Since:
3.0
  • Field Details

    • lambda

      private int lambda
      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.
    • isActiveCMA

      private final boolean isActiveCMA
      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.
    • checkFeasableCount

      private final int checkFeasableCount
      Determines how often a new random offspring is generated in case it is not feasible / beyond the defined limits, default is 0.
    • inputSigma

      private double[] inputSigma
      See Also:
    • dimension

      private int dimension
      Number of objective variables/problem dimension
    • diagonalOnly

      private int diagonalOnly
      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 Also:
    • isMinimize

      private boolean isMinimize
      Number of objective variables/problem dimension
    • generateStatistics

      private final boolean generateStatistics
      Indicates whether statistic data is collected.
    • maxIterations

      private final int maxIterations
      Maximal number of iterations allowed.
    • stopFitness

      private final double stopFitness
      Limit for fitness value.
    • stopTolUpX

      private double stopTolUpX
      Stop if x-changes larger stopTolUpX.
    • stopTolX

      private double stopTolX
      Stop if x-change smaller stopTolX.
    • stopTolFun

      private double stopTolFun
      Stop if fun-changes smaller stopTolFun.
    • stopTolHistFun

      private double stopTolHistFun
      Stop if back fun-changes smaller stopTolHistFun.
    • mu

      private int mu
      Number of parents/points for recombination.
    • logMu2

      private double logMu2
      log(mu + 0.5), stored for efficiency.
    • weights

      private RealMatrix weights
      Array for weighted recombination.
    • mueff

      private double mueff
      Variance-effectiveness of sum w_i x_i.
    • sigma

      private double sigma
      Overall standard deviation - search volume.
    • cc

      private double cc
      Cumulation constant.
    • cs

      private double cs
      Cumulation constant for step-size.
    • damps

      private double damps
      Damping for step-size.
    • ccov1

      private double ccov1
      Learning rate for rank-one update.
    • ccovmu

      private double ccovmu
      Learning rate for rank-mu update'
    • chiN

      private double chiN
      Expectation of ||N(0,I)|| == norm(randn(N,1)).
    • ccov1Sep

      private double ccov1Sep
      Learning rate for rank-one update - diagonalOnly
    • ccovmuSep

      private double ccovmuSep
      Learning rate for rank-mu update - diagonalOnly
    • xmean

      private RealMatrix xmean
      Objective variables.
    • pc

      private RealMatrix pc
      Evolution path.
    • ps

      private RealMatrix ps
      Evolution path for sigma.
    • normps

      private double normps
      Norm of ps, stored for efficiency.
    • B

      private RealMatrix B
      Coordinate system.
    • D

      private RealMatrix D
      Scaling.
    • BD

      private RealMatrix BD
      B*D, stored for efficiency.
    • diagD

      private RealMatrix diagD
      Diagonal of sqrt(D), stored for efficiency.
    • C

      private RealMatrix C
      Covariance matrix.
    • diagC

      private RealMatrix diagC
      Diagonal of C, used for diagonalOnly.
    • iterations

      private int iterations
      Number of iterations already performed.
    • fitnessHistory

      private double[] fitnessHistory
      History queue of best values.
    • historySize

      private int historySize
      Size of history queue of best values.
    • random

      private final RandomGenerator random
      Random generator.
    • statisticsSigmaHistory

      private final List<Double> statisticsSigmaHistory
      History of sigma values.
    • statisticsMeanHistory

      private final List<RealMatrix> statisticsMeanHistory
      History of mean matrix.
    • statisticsFitnessHistory

      private final List<Double> statisticsFitnessHistory
      History of fitness values.
    • statisticsDHistory

      private final List<RealMatrix> statisticsDHistory
      History of D matrix.
  • Constructor Details

    • CMAESOptimizer

      public CMAESOptimizer(int maxIterations, double stopFitness, boolean isActiveCMA, int diagonalOnly, int checkFeasableCount, RandomGenerator random, boolean generateStatistics, ConvergenceChecker<PointValuePair> checker)
      Parameters:
      maxIterations - Maximal number of iterations.
      stopFitness - Whether to stop if objective function value is smaller than stopFitness.
      isActiveCMA - Chooses the covariance matrix update method.
      diagonalOnly - Number of initial iterations, where the covariance matrix remains diagonal.
      checkFeasableCount - Determines how often new random objective variables are generated in case they are out of bounds.
      random - Random generator.
      generateStatistics - Whether statistic data is collected.
      checker - Convergence checker.
      Since:
      3.1
  • Method Details

    • getStatisticsSigmaHistory

      public List<Double> getStatisticsSigmaHistory()
      Returns:
      History of sigma values.
    • getStatisticsMeanHistory

      public List<RealMatrix> getStatisticsMeanHistory()
      Returns:
      History of mean matrix.
    • getStatisticsFitnessHistory

      public List<Double> getStatisticsFitnessHistory()
      Returns:
      History of fitness values.
    • getStatisticsDHistory

      public List<RealMatrix> getStatisticsDHistory()
      Returns:
      History of D matrix.
    • optimize

      Stores data and performs the optimization.

      The list of parameters is open-ended so that sub-classes can extend it with arguments specific to their concrete implementations.

      When the method is called multiple times, instance data is overwritten only when actually present in the list of arguments: when not specified, data set in a previous call is retained (and thus is optional in subsequent calls).

      Important note: Subclasses must override BaseOptimizer.parseOptimizationData(OptimizationData[]) if they need to register their own options; but then, they must also call super.parseOptimizationData(optData) within that method.

      Overrides:
      optimize in class MultivariateOptimizer
      Parameters:
      optData - Optimization data. In addition to those documented in MultivariateOptimizer, this method will register the following data:
      Returns:
      a point/value pair that satisfies the convergence criteria.
      Throws:
      TooManyEvaluationsException - if the maximal number of evaluations is exceeded.
      DimensionMismatchException - if the initial guess, target, and weight arguments have inconsistent dimensions.
    • doOptimize

      protected PointValuePair doOptimize()
      Performs the bulk of the optimization algorithm.
      Specified by:
      doOptimize in class BaseOptimizer<PointValuePair>
      Returns:
      the point/value pair giving the optimal value of the objective function.
    • parseOptimizationData

      protected void parseOptimizationData(OptimizationData... optData)
      Scans the list of (required and optional) optimization data that characterize the problem.
      Overrides:
      parseOptimizationData in class MultivariateOptimizer
      Parameters:
      optData - Optimization data. The following data will be looked for:
    • checkParameters

      private void checkParameters()
      Checks dimensions and values of boundaries and inputSigma if defined.
    • initializeCMA

      private void initializeCMA(double[] guess)
      Initialization of the dynamic search parameters
      Parameters:
      guess - Initial guess for the arguments of the fitness function.
    • updateEvolutionPaths

      private boolean updateEvolutionPaths(RealMatrix zmean, RealMatrix xold)
      Update of the evolution paths ps and pc.
      Parameters:
      zmean - Weighted row matrix of the gaussian random numbers generating the current offspring.
      xold - xmean matrix of the previous generation.
      Returns:
      hsig flag indicating a small correction.
    • updateCovarianceDiagonalOnly

      private void updateCovarianceDiagonalOnly(boolean hsig, RealMatrix bestArz)
      Update of the covariance matrix C for diagonalOnly > 0
      Parameters:
      hsig - Flag indicating a small correction.
      bestArz - Fitness-sorted matrix of the gaussian random values of the current offspring.
    • updateCovariance

      private void updateCovariance(boolean hsig, RealMatrix bestArx, RealMatrix arz, int[] arindex, RealMatrix xold)
      Update of the covariance matrix C.
      Parameters:
      hsig - Flag indicating a small correction.
      bestArx - Fitness-sorted matrix of the argument vectors producing the current offspring.
      arz - Unsorted matrix containing the gaussian random values of the current offspring.
      arindex - Indices indicating the fitness-order of the current offspring.
      xold - xmean matrix of the previous generation.
    • updateBD

      private void updateBD(double negccov)
      Update B and D from C.
      Parameters:
      negccov - Negative covariance factor.
    • push

      private static void push(double[] vals, double val)
      Pushes the current best fitness value in a history queue.
      Parameters:
      vals - History queue.
      val - Current best fitness value.
    • sortedIndices

      private int[] sortedIndices(double[] doubles)
      Sorts fitness values.
      Parameters:
      doubles - Array of values to be sorted.
      Returns:
      a sorted array of indices pointing into doubles.
    • valueRange

      private double valueRange(CMAESOptimizer.ValuePenaltyPair[] vpPairs)
      Get range of values.
      Parameters:
      vpPairs - Array of valuePenaltyPairs to get range from.
      Returns:
      a double equal to maximum value minus minimum value.
    • log

      private static RealMatrix log(RealMatrix m)
      Parameters:
      m - Input matrix
      Returns:
      Matrix representing the element-wise logarithm of m.
    • sqrt

      private static RealMatrix sqrt(RealMatrix m)
      Parameters:
      m - Input matrix.
      Returns:
      Matrix representing the element-wise square root of m.
    • square

      private static RealMatrix square(RealMatrix m)
      Parameters:
      m - Input matrix.
      Returns:
      Matrix representing the element-wise square of m.
    • times

      private static RealMatrix times(RealMatrix m, RealMatrix n)
      Parameters:
      m - Input matrix 1.
      n - Input matrix 2.
      Returns:
      the matrix where the elements of m and n are element-wise multiplied.
    • divide

      private static RealMatrix divide(RealMatrix m, RealMatrix n)
      Parameters:
      m - Input matrix 1.
      n - Input matrix 2.
      Returns:
      Matrix where the elements of m and n are element-wise divided.
    • selectColumns

      private static RealMatrix selectColumns(RealMatrix m, int[] cols)
      Parameters:
      m - Input matrix.
      cols - Columns to select.
      Returns:
      Matrix representing the selected columns.
    • triu

      private static RealMatrix triu(RealMatrix m, int k)
      Parameters:
      m - Input matrix.
      k - Diagonal position.
      Returns:
      Upper triangular part of matrix.
    • sumRows

      private static RealMatrix sumRows(RealMatrix m)
      Parameters:
      m - Input matrix.
      Returns:
      Row matrix representing the sums of the rows.
    • diag

      private static RealMatrix diag(RealMatrix m)
      Parameters:
      m - Input matrix.
      Returns:
      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.
    • copyColumn

      private static void copyColumn(RealMatrix m1, int col1, RealMatrix m2, int col2)
      Copies a column from m1 to m2.
      Parameters:
      m1 - Source matrix.
      col1 - Source column.
      m2 - Target matrix.
      col2 - Target column.
    • ones

      private static RealMatrix ones(int n, int m)
      Parameters:
      n - Number of rows.
      m - Number of columns.
      Returns:
      n-by-m matrix filled with 1.
    • eye

      private static RealMatrix eye(int n, int m)
      Parameters:
      n - Number of rows.
      m - Number of columns.
      Returns:
      n-by-m matrix of 0 values out of diagonal, and 1 values on the diagonal.
    • zeros

      private static RealMatrix zeros(int n, int m)
      Parameters:
      n - Number of rows.
      m - Number of columns.
      Returns:
      n-by-m matrix of zero values.
    • repmat

      private static RealMatrix repmat(RealMatrix mat, int n, int m)
      Parameters:
      mat - Input matrix.
      n - Number of row replicates.
      m - Number of column replicates.
      Returns:
      a matrix which replicates the input matrix in both directions.
    • sequence

      private static RealMatrix sequence(double start, double end, double step)
      Parameters:
      start - Start value.
      end - End value.
      step - Step size.
      Returns:
      a sequence as column matrix.
    • max

      private static double max(RealMatrix m)
      Parameters:
      m - Input matrix.
      Returns:
      the maximum of the matrix element values.
    • min

      private static double min(RealMatrix m)
      Parameters:
      m - Input matrix.
      Returns:
      the minimum of the matrix element values.
    • max

      private static double max(double[] m)
      Parameters:
      m - Input array.
      Returns:
      the maximum of the array values.
    • min

      private static double min(double[] m)
      Parameters:
      m - Input array.
      Returns:
      the minimum of the array values.
    • inverse

      private static int[] inverse(int[] indices)
      Parameters:
      indices - Input index array.
      Returns:
      the inverse of the mapping defined by indices.
    • reverse

      private static int[] reverse(int[] indices)
      Parameters:
      indices - Input index array.
      Returns:
      the indices in inverse order (last is first).
    • randn

      private double[] randn(int size)
      Parameters:
      size - Length of random array.
      Returns:
      an array of Gaussian random numbers.
    • randn1

      private RealMatrix randn1(int size, int popSize)
      Parameters:
      size - Number of rows.
      popSize - Population size.
      Returns:
      a 2-dimensional matrix of Gaussian random numbers.