# Linear Covariance Matrix Adaptation (VDCMA)¶

class pypop7.optimizers.es.vdcma.VDCMA(problem, options)

Parameters:
• 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)).

Examples

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 pypop7.optimizers.es.vdcma 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.

mean

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

Type:

array_like

n_individuals

number of offspring, aka offspring population size.

Type:

int

n_parents

number of parents, aka parental population size.

Type:

int

sigma

final global step-size, aka mutation strength.

Type:

float

References

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. https://dl.acm.org/doi/abs/10.1145/2576768.2598258

See the official Python version from Prof. Akimoto: https://gist.github.com/youheiakimoto/08b95b52dfbf8832afc71dfff3aed6c8