Linear Covariance Matrix Adaptation (VDCMA)

class, options)

Linear Covariance Matrix Adaptation (VDCMA).

  • problem (dict) –

    problem arguments with the following common settings (keys):
    • ’fitness_function’ - objective function to be minimized (func),

    • ’ndim_problem’ - number of dimensionality (int),

    • ’upper_boundary’ - upper boundary of search range (array_like),

    • ’lower_boundary’ - lower boundary of search range (array_like).

  • options (dict) –

    optimizer options with the following common settings (keys):
    • ’max_function_evaluations’ - maximum of function evaluations (int, default: np.Inf),

    • ’max_runtime’ - maximal runtime to be allowed (float, default: np.Inf),

    • ’seed_rng’ - seed for random number generation needed to be explicitly set (int);

    and with the following particular settings (keys):
    • ’sigma’ - initial global step-size, aka mutation strength (float),

    • ’mean’ - initial (starting) point, aka mean of Gaussian search distribution (array_like),

      • if not given, it will draw a random sample from the uniform distribution whose search range is bounded by problem[‘lower_boundary’] and problem[‘upper_boundary’].

    • ’n_individuals’ - number of offspring, aka offspring population size (int, default: 4 + int(3*np.log(problem[‘ndim_problem’]))),

    • ’n_parents’ - number of parents, aka parental population size (int, default: int(options[‘n_individuals’]/2)).


Use the optimizer to minimize the well-known test function Rosenbrock:

 1>>> import numpy
 2>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
 3>>> from import VDCMA
 4>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
 5...            'ndim_problem': 2,
 6...            'lower_boundary': -5*numpy.ones((2,)),
 7...            'upper_boundary': 5*numpy.ones((2,))}
 8>>> options = {'max_function_evaluations': 5000,  # set optimizer options
 9...            'seed_rng': 2022,
10...            'mean': 3*numpy.ones((2,)),
11...            'sigma': 0.1}  # the global step-size may need to be tuned for better performance
12>>> vdcma = VDCMA(problem, options)  # initialize the optimizer class
13>>> results = vdcma.optimize()  # run the optimization process
14>>> # return the number of function evaluations and best-so-far fitness
15>>> print(f"VDCMA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
16VDCMA: 5000, 7.116226375179302e-18

For its correctness checking of coding, refer to this code-based repeatability report for more details.


initial (starting) point, aka mean of Gaussian search distribution.




number of offspring, aka offspring population size.




number of parents, aka parental population size.




final global step-size, aka mutation strength.




Akimoto, Y., Auger, A. and Hansen, N., 2014, July. Comparison-based natural gradient optimization in high dimension. In Proceedings of Annual Conference on Genetic and Evolutionary Computation (pp. 373-380). ACM.

See the official Python version from Prof. Akimoto: