scipy.optimize.differential_evolution

scipy.optimize.differential_evolution(func, bounds, args=(), strategy='best1bin', maxiter=1000, popsize=15, tol=0.01, mutation=0.5, 1, recombination=0.7, seed=None, callback=None, disp=False, polish=True, init='latinhypercube', atol=0, updating='immediate', workers=1, constraints=())

Finds the global minimum of a multivariate function.

Differential Evolution is stochastic in nature (does not use gradient methods) to find the minimum, and can search large areas of candidate space, but often requires larger numbers of function evaluations than conventional gradient-based techniques.

The algorithm is due to Storn and Price [1].

Parameters

funccallable

The objective function to be minimized. Must be in the form f(x, *args), where x is the argument in the form of a 1-D array and args is a tuple of any additional fixed parameters needed to completely specify the function.

boundssequence or Bounds, optional

Bounds for variables. There are two ways to specify the bounds: 1. Instance of Bounds class. 2. (min, max) pairs for each element in x, defining the finite lower and upper bounds for the optimizing argument of func. It is required to have len(bounds) == len(x). len(bounds) is used to determine the number of parameters in x.

argstuple, optional

Any additional fixed parameters needed to completely specify the objective function.

strategystr, optional

The differential evolution strategy to use. Should be one of:

• ‘best1bin’

• ‘best1exp’

• ‘rand1exp’

• ‘randtobest1exp’

• ‘currenttobest1exp’

• ‘best2exp’

• ‘rand2exp’

• ‘randtobest1bin’

• ‘currenttobest1bin’

• ‘best2bin’

• ‘rand2bin’

• ‘rand1bin’

The default is ‘best1bin’.

maxiterint, optional

The maximum number of generations over which the entire population is evolved. The maximum number of function evaluations (with no polishing) is: (maxiter + 1) * popsize * len(x)

popsizeint, optional

A multiplier for setting the total population size. The population has popsize * len(x) individuals (unless the initial population is supplied via the init keyword).

tolfloat, optional

Relative tolerance for convergence, the solving stops when np.std(pop) <= atol + tol * np.abs(np.mean(population_energies)), where and atol and tol are the absolute and relative tolerance respectively.

mutationfloat or tuple(float, float), optional

The mutation constant. In the literature this is also known as differential weight, being denoted by F. If specified as a float it should be in the range [0, 2]. If specified as a tuple (min, max) dithering is employed. Dithering randomly changes the mutation constant on a generation by generation basis. The mutation constant for that generation is taken from U[min, max). Dithering can help speed convergence significantly. Increasing the mutation constant increases the search radius, but will slow down convergence.

recombinationfloat, optional

The recombination constant, should be in the range [0, 1]. In the literature this is also known as the crossover probability, being denoted by CR. Increasing this value allows a larger number of mutants to progress into the next generation, but at the risk of population stability.

seed{int, RandomState, Generator}, optional

If seed is not specified the RandomState singleton is used. If seed is an int, a new RandomState instance is used, seeded with seed. If seed is already a RandomState or a Generator instance, then that object is used. Specify seed for repeatable minimizations.

dispbool, optional

Prints the evaluated func at every iteration.

callbackcallable, callback(xk, convergence=val), optional

A function to follow the progress of the minimization. xk is the current value of x0. val represents the fractional value of the population convergence. When val is greater than one the function halts. If callback returns True, then the minimization is halted (any polishing is still carried out).

polishbool, optional

If True (default), then scipy.optimize.minimize with the L-BFGS-B method is used to polish the best population member at the end, which can improve the minimization slightly. If a constrained problem is being studied then the trust-constr method is used instead.

initstr or array-like, optional

Specify which type of population initialization is performed. Should be one of:

• ‘latinhypercube’

• ‘random’

• array specifying the initial population. The array should have shape (M, len(x)), where M is the total population size and len(x) is the number of parameters. init is clipped to bounds before use.

The default is ‘latinhypercube’. Latin Hypercube sampling tries to maximize coverage of the available parameter space. ‘random’ initializes the population randomly - this has the drawback that clustering can occur, preventing the whole of parameter space being covered. Use of an array to specify a population subset could be used, for example, to create a tight bunch of initial guesses in an location where the solution is known to exist, thereby reducing time for convergence.

atolfloat, optional

Absolute tolerance for convergence, the solving stops when np.std(pop) <= atol + tol * np.abs(np.mean(population_energies)), where and atol and tol are the absolute and relative tolerance respectively.

updating{‘immediate’, ‘deferred’}, optional

If 'immediate', the best solution vector is continuously updated within a single generation [4]. This can lead to faster convergence as trial vectors can take advantage of continuous improvements in the best solution. With 'deferred', the best solution vector is updated once per generation. Only 'deferred' is compatible with parallelization, and the workers keyword can over-ride this option.

New in version 1.2.0.

workersint or map-like callable, optional

If workers is an int the population is subdivided into workers sections and evaluated in parallel (uses multiprocessing.Pool). Supply -1 to use all available CPU cores. Alternatively supply a map-like callable, such as multiprocessing.Pool.map for evaluating the population in parallel. This evaluation is carried out as workers(func, iterable). This option will override the updating keyword to updating='deferred' if workers != 1. Requires that func be pickleable.

New in version 1.2.0.

constraints{NonLinearConstraint, LinearConstraint, Bounds}

Constraints on the solver, over and above those applied by the bounds kwd. Uses the approach by Lampinen [5].

New in version 1.4.0.

Returns

resOptimizeResult

The optimization result represented as a OptimizeResult object. Important attributes are: x the solution array, success a Boolean flag indicating if the optimizer exited successfully and message which describes the cause of the termination. See OptimizeResult for a description of other attributes. If polish was employed, and a lower minimum was obtained by the polishing, then OptimizeResult also contains the jac attribute. If the eventual solution does not satisfy the applied constraints success will be False.

Notes

Differential evolution is a stochastic population based method that is useful for global optimization problems. At each pass through the population the algorithm mutates each candidate solution by mixing with other candidate solutions to create a trial candidate. There are several strategies [2] for creating trial candidates, which suit some problems more than others. The ‘best1bin’ strategy is a good starting point for many systems. In this strategy two members of the population are randomly chosen. Their difference is used to mutate the best member (the ‘best’ in ‘best1bin’), \(b_0\), so far:

\[b' = b_0 + mutation * (population[rand0] - population[rand1])\]

A trial vector is then constructed. Starting with a randomly chosen ith parameter the trial is sequentially filled (in modulo) with parameters from b' or the original candidate. The choice of whether to use b' or the original candidate is made with a binomial distribution (the ‘bin’ in ‘best1bin’) - a random number in [0, 1) is generated. If this number is less than the recombination constant then the parameter is loaded from b', otherwise it is loaded from the original candidate. The final parameter is always loaded from b'. Once the trial candidate is built its fitness is assessed. If the trial is better than the original candidate then it takes its place. If it is also better than the best overall candidate it also replaces that. To improve your chances of finding a global minimum use higher popsize values, with higher mutation and (dithering), but lower recombination values. This has the effect of widening the search radius, but slowing convergence. By default the best solution vector is updated continuously within a single iteration (updating='immediate'). This is a modification [4] of the original differential evolution algorithm which can lead to faster convergence as trial vectors can immediately benefit from improved solutions. To use the original Storn and Price behaviour, updating the best solution once per iteration, set updating='deferred'.

New in version 0.15.0.

References

1

Storn, R and Price, K, Differential Evolution - a Simple and Efficient Heuristic for Global Optimization over Continuous Spaces, Journal of Global Optimization, 1997, 11, 341 - 359.

2

http://www1.icsi.berkeley.edu/~storn/code.html

3

http://en.wikipedia.org/wiki/Differential_evolution

4(1,2)

Wormington, M., Panaccione, C., Matney, K. M., Bowen, D. K., - Characterization of structures from X-ray scattering data using genetic algorithms, Phil. Trans. R. Soc. Lond. A, 1999, 357, 2827-2848

5

Lampinen, J., A constraint handling approach for the differential evolution algorithm. Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02 (Cat. No. 02TH8600). Vol. 2. IEEE, 2002.

Examples

Let us consider the problem of minimizing the Rosenbrock function. This function is implemented in rosen in scipy.optimize.

>>> from scipy.optimize import rosen, differential_evolution
>>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
>>> result = differential_evolution(rosen, bounds)
>>> result.x, result.fun
(array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19)

Now repeat, but with parallelization.

>>> bounds = [(0,2), (0, 2), (0, 2), (0, 2), (0, 2)]
>>> result = differential_evolution(rosen, bounds, updating='deferred',
...                                 workers=2)
>>> result.x, result.fun
(array([1., 1., 1., 1., 1.]), 1.9216496320061384e-19)

Let’s try and do a constrained minimization

>>> from scipy.optimize import NonlinearConstraint, Bounds
>>> def constr_f(x):
...     return np.array(x[0] + x[1])
>>>
>>> # the sum of x[0] and x[1] must be less than 1.9
>>> nlc = NonlinearConstraint(constr_f, -np.inf, 1.9)
>>> # specify limits using a `Bounds` object.
>>> bounds = Bounds([0., 0.], [2., 2.])
>>> result = differential_evolution(rosen, bounds, constraints=(nlc),
...                                 seed=1)
>>> result.x, result.fun
(array([0.96633867, 0.93363577]), 0.0011361355854792312)

Next find the minimum of the Ackley function (https://en.wikipedia.org/wiki/Test_functions_for_optimization).

>>> from scipy.optimize import differential_evolution
>>> import numpy as np
>>> def ackley(x):
...     arg1 = -0.2 * np.sqrt(0.5 * (x[0] ** 2 + x[1] ** 2))
...     arg2 = 0.5 * (np.cos(2. * np.pi * x[0]) + np.cos(2. * np.pi * x[1]))
...     return -20. * np.exp(arg1) - np.exp(arg2) + 20. + np.e
>>> bounds = [(-5, 5), (-5, 5)]
>>> result = differential_evolution(ackley, bounds)
>>> result.x, result.fun
(array([ 0.,  0.]), 4.4408920985006262e-16)