# Univariate Marginal Distribution Algorithm (UMDA)¶

class pypop7.optimizers.eda.umda.UMDA(problem, options)

Univariate Marginal Distribution Algorithm for normal models (UMDA).

Note

UMDA learns only the diagonal elements of covariance matrix of the Gaussian sampling distribution, resulting in a linear time complexity w.r.t. each sampling. Therefore, it can be seen as a baseline for large-scale black-box optimization (LSBBO). To obtain satisfactory performance for LSBBO, the number of offspring may need to be carefully tuned.

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 (float, default: np.Inf),

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

and with the following particular settings (keys):
• ’n_individuals’ - number of offspring, aka offspring population size (int, default: 200),

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

Examples

Use the optimizer UMDA 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.eda.umda import UMDA
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>>> umda = UMDA(problem, options)  # initialize the optimizer class
11>>> results = umda.optimize()  # run the optimization process
12>>> # return the number of function evaluations and best-so-far fitness
13>>> print(f"UMDA: {results['n_function_evaluations']}, {results['best_so_far_y']}")
14UMDA: 5000, 0.029323401402499186
```

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

n_individuals

number of offspring, aka offspring population size.

Type:

int

n_parents

number of parents, aka parental population size.

Type:

int

References

Mühlenbein, H. and Mahnig, T., 2002. Evolutionary computation and Wright’s equation. Theoretical Computer Science, 287(1), pp.145-165. https://www.sciencedirect.com/science/article/pii/S0304397502000981

Larrañaga, P. and Lozano, J.A. eds., 2001. Estimation of distribution algorithms: A new tool for evolutionary computation. Springer Science & Business Media. https://link.springer.com/book/10.1007/978-1-4615-1539-5

Mühlenbein, H. and Mahnig, T., 2001. Evolutionary algorithms: From recombination to search distributions. In Theoretical Aspects of Evolutionary Computing (pp. 135-173). Springer, Berlin, Heidelberg. https://link.springer.com/chapter/10.1007/978-3-662-04448-3_7

Larranaga, P., Etxeberria, R., Lozano, J.A. and Pena, J.M., 2000. Optimization in continuous domains by learning and simulation of Gaussian networks. Technical Report, Department of Computer Science and Artificial Intelligence, University of the Basque Country. https://tinyurl.com/3bw6n3x4

Larranaga, P., Etxeberria, R., Lozano, J.A. and Pe, J.M., 1999. Optimization by learning and simulation of Bayesian and Gaussian networks. Technical Report, Department of Computer Science and Artificial Intelligence, University of the Basque Country. https://tinyurl.com/5dktrdwc

Mühlenbein, H., 1997. The equation for response to selection and its use for prediction. Evolutionary Computation, 5(3), pp.303-346. https://tinyurl.com/yt78c786