Generalized Pattern Search (GPS)

class pypop7.optimizers.ds.gps.GPS(problem, options)

Generalized Pattern Search (GPS).

Note

“To converge to a local minimum, certain conditions must be met. The set of directions must be a positive spanning set, which means that we can construct any point using a nonnegative linear combination of the directions. A positive spanning set ensures that at least one of the directions is a descent direction from a location with a nonzero gradient.”—[Kochenderfer&Wheeler, 2019]

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 (float, default: 1.0),

  • ’x’ - initial (starting) point (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’].

  • ’gamma’ - decreasing factor of step-size (float, default: 0.5).

Examples

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

>>> import numpy
>>> from pypop7.benchmarks.base_functions import rosenbrock  # function to be minimized
>>> from pypop7.optimizers.ds.gps import GPS
>>> problem = {'fitness_function': rosenbrock,  # define problem arguments
...            'ndim_problem': 2,
...            'lower_boundary': -5*numpy.ones((2,)),
...            'upper_boundary': 5*numpy.ones((2,))}
>>> options = {'max_function_evaluations': 5000,  # set optimizer options
...            'seed_rng': 2022,
...            'x': 3*numpy.ones((2,)),
...            'sigma': 0.1,
...            'verbose_frequency': 500}
>>> gps = GPS(problem, options)  # initialize the optimizer class
>>> results = gps.optimize()  # run the optimization process
>>> # return the number of function evaluations and best-so-far fitness
>>> print(f"GPS: {results['n_function_evaluations']}, {results['best_so_far_y']}")
GPS: 5000, 0.6182686369768672

gamma

decreasing factor of step-size.

Type:

float

sigma

final global step-size (changed during optimization).

Type:

float

x

initial (starting) point.

Type:

array_like

References

Kochenderfer, M.J. and Wheeler, T.A., 2019. Algorithms for optimization. MIT Press. https://algorithmsbook.com/optimization/files/chapter-7.pdf (See Algorithm 7.6 (Page 106) for details.)

Regis, R.G., 2016. On the properties of positive spanning sets and positive bases. Optimization and Engineering, 17(1), pp.229-262. https://link.springer.com/article/10.1007/s11081-015-9286-x

Torczon, V., 1997. On the convergence of pattern search algorithms. SIAM Journal on Optimization, 7(1), pp.1-25. https://epubs.siam.org/doi/abs/10.1137/S1052623493250780